Divide, Harmonize, Then Conquer It: Shooting Multi-Commodity Flow Problems with Multimodal Language Models
Authors: Xinyu Yuan
Yan Qiao
Zonghui Wang
Wenzhi CHEN
Venue: ICLR 2026 Poster (2026)
Keywords: Multi-Commodity Flow
Multimodal Language Models
Resource Allocation
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: Explore the usage of MLMs for solving multi-commodity flow problems quickly

Abstract: The multi-commodity flow (MCF) problem is a fundamental topic in network flow and combinatorial optimization, with broad applications in transportation, communication, and logistics, etc. Nowadays, the rapid expansion of allocation systems has posed challenges for existing optimization engines in balancing optimality and tractability. In this paper, we present Pram, the first ML-based method that leverages the reasoning power of multimodal language models (MLMs) for addressing the trade-off dilemma—a great need of service providers. As part of our proposal, Pram (i) quickly computes high-quality allocations by dividing the original problem into local subproblems, which are then resolved by an MLM-powered "agent", and (ii) ensures global consistency by harmonizing these subproblems via a multi-agent reinforcement learning algorithm. Theoretically, we show that Pram, which learns to perform gradient descent in context, provably converges to the optimum within the family of MCF problems. 
Empirically, on real-world datasets and public topologies, Pram achieves performance comparable to, and in some cases even surpassing, linear programming solvers (very close to the optimal solution), and substantially lower runtimes (one to two orders of magnitude faster). 
Moreover, Pram exhibits strong robustness (<10\% performance degradation under failures or bursts), demonstrating MLM's generalization ability to unforeseen events. 
Pram  is objective-agnostic and seamlessly integrates with mainstream allocation systems, providing a practical and scalable solution for future networks.

Published as a conference paper at ICLR 2026
DIVIDE, HARMONIZE, THEN CONQUER IT: SHOOTING MULTI-COMMODITY FLOW PROBLEMS WITH MULTI- MODAL LANGUAGE MODELS*
Xinyu Yuan Yan Qiao v. Zonghui Wang Wenzhi Chen
A ZZejianng University Hefei University Technology "Co-corresponding authors yxy53158gmain com, qiacyan&hfut. eeooyaanffuueddu.....
edu. cn, w {zhwang, chenwz ) jzju. edu. on
ABSTRACT
The multi-commodity flow (MCF) problem fundamental topic in network flow and combinatorial optimization, with broad applications in transportation, communication, and logistics, etc. Nowadays, the rapid expansion of allocation systems has posed challenges for existing optimization engines in balancing opti- mality and tractability. In this paper, we present PRAM, the first ML-based method that leverages the reasoning power of multimodal language models (MLMs) for addressing the trade-off dilemma- great need of service providers. As part of our proposal, PRAM (i) quickly computes high-quality allocations by dividing the orig- inal problem into local subproblems, which are then resolved by an --W-prwered "agent", and (ii) ensures global consistency by harmonizing these subproblems via a multi-agent reinforcement learning algorithm. Theoretically, we show that PRAM, which learns to perform gradient descent in context, provably converges to the optimum within the family of MCF problems. Empirically, on real-world datasets and public topologies, PRAM achieves performance comparable to, and in some cases even surpassing, linear programming solvers (very close to the optimal solution), and substantially lower runtimes (1~2 orders of magnitude faster). More- over, PRAM exhibits strong robustness (<10% performance degradation under link failures or flow bursts), demonstrating MLM's generalization ability to unforeseen events. PRAM is objective-agnostic and seamlessly integrates with mainstream allocation systems, providing a practical and scalable solution for future networks.
INTRODUCTION
Suppose we want to send multiple commodities from their sources to their respective destinations along the ares of underlying network, with the objectives of achieving low link utilization, high throughput, and fairness among commodities. This scenario results in what we call a multi- commodity flow (MCF) problem (Assad, 1978). Its importance has been underscored by wide-ranging applications in transportation, communication, logistics, energy, and cloud computing (Schrijver, 2002; Wang & Wang, 1999: Balcik al., 2014: Blaauwbrock et al., 2015; Chang et al., 2010). Over the past decades, optimization- based algorithms built on linear programming (LP) played an important role in solving such problems, which is guaranteed for computing near-optimal solutions (Khachiyan, 1980; Karmarkar, 1984; Chen & Ye, 2024). However, the conventional wisdom in the community is that solving these LPss becomes time-consuming as the solution space scales up (Cohen et al., 2021). This bottleneck is further magnified in modern systems, which often comprise thousands of nodes or/and links, and must accommodate millions commodities with unpredictable demands (Applegate & Cohen, 2003; Wang al., 2006). More recently, the advancements of machine learning (ML) have prompted extensive research into ML-based solutions (Valadarsky et al., 2017; Bernárdez et al., 2021). While faster, these methods are still limited by the problem scale ($ 3.1), the number of variables to be optimized remains unchanged. Additionally, they inherit the common shortcomings of MI. models, e.g., sensitivity unseen environments and painstaking hyperparameter sweeping.
In response to the central barrier of scalability, we take different perspective. Rather than monolithic optimization, our primary idea is handle the growing network size by decomposing the original
OOr implementation is publicly available ? itttpp://gi thub. com/ Boeonn/ddebuuustee sys/ PPAM.

---

Published as a conference paper at ICLR 2026
Network Communication
Transportation System
Logishics Scheduling
Power Transmission
Cloud Computingg
Figure 1: Various real-world examples of multi-commodity flow. From left to right: wide-ares network traffic engineering, urban mobility management, delivery route optimization, regional power dispatch, and tenant-aware flow control, all of which involve a very large solution space today.
problem into smaller subproblems through the partition of topology and demands (Abuzaid et al., 2021; Cohen et al., 2021), and given historical data, directly solving them in parallel through deep neural networks (DNNs). Building on this idea, we propose PRAM: Partitioned Resource Allocation with Multimodal Language Models (MLMs). First, we divide the MCF problem by commodity flow sources, then allocate flows for each subset individually using a shared "agent" model. We here choose MLMs as the agent backbone for two major reasons: I they exhibit emergent abilities that were not explicitly programmed into them during pre-training on massive data, such as mathematical problem reasoning and generalization to unforeseen conditions (Zhang et al., 2024; Wu et al., 2024a); and 2 they mitigate the costs of retraining and handcrafting specialized DNNs for complex inputs, by processing topology and demand data images and text tokens, respectively. Second, present a novel multi-agent reinforcement learning (MARL) algorithm for fine-tuning the agent model using counterfactual policy gradients (Foerster et al., 2018), with lightweight communication enabled by trainable low-rank matrices and prefix context. This allows each logical agent to exchange and estimate its individual contribution the team's 's success. Further, through case studies, we show that since the MCF objective typically satisfies suitable convexity/concavity properties, once adapted, PRAM provably attains the optimum by internally simulating the gradient descent (GD) procedure. Hence, the effectiveness of PRAM rests on a solid theoretical foundation. To this end, the paper makes three key contributions which can be summarized as follows:
In remarkable departure from conventional centralized consensus, we propose PRAM ($ 3) to divide and optimize MCF problems in a distributed manner, thereby providing efficiency at scale.
PRAM is the first end-to-end MCF solver built on off-the-sheit MLMs ($ 3.2), with no redundant manual design overhead. It employs a lightweight and interpretable adaptation framework ($ 3.3) that counters partition effects and fully taps the model's 's reasoning for different flow allocation tasks.
A suite of theoretical results ($ 4) demonstrates that, once adapted, PRAM can internally approximate near-optimal solutions, offering performance guarantees largely absent in prior works.
Our evaluation ($ 5) encompasses topologies of different scales, heterogeneous demand distributions, and multiple MCF objectives. In comparison with a variety of solutions, including RL, LP and heuristics, the main results show that: ( PRAM achieves near-optimal performance. It consistently outperforms previous RL-based allocation schemes, with an average performance gap of less than 8% from the optimal solution. 2 PRAM accelerates flow allocation at scale. It speeds 10x to 100x faster than solving LPs on large-scale topologies. @ PRAM generalizes well to new environment. It exhibits strong robustness to demand distributions, flow dynamics, and network failures. Through ablation studies and visualizations, we delve deeper into the factors contributing to PRAM'S effectiveness.
In conclusion ($ 6), we regard our exploration of MLM--ooered MCF optimization as an initial yet groundbreaking step, and discuss the limitations of PRAM that we hope future research can address.
2 PROBLEM FORMULATION
We formulate the multi-commodii flow (MCF) problem on directed graph or topology G(V, & c). where V is the set of vertices or nodes, & the set directed edges or links, and c: E + R+ specifies link capacities. Each source-destination pair s, t is associated with a predefined set of candidate paths Pa. and historical demand Da,t- A configuration π distributes each commodity flow using candidate path weights Tp. The goal is to periodically determine the optimal under different objectives, e.g., minimizing maximum link utilization for resilience, maximizing total flow for profit, or concurrent flow for faimess (see Appendix B.1 for objective details). This problem is known be
2

---

Published as a conference paper at ICLR 2026
oub images
Shared Agent
Representch ona
Agent-Dacntin Enoccdiing
Weihtt
Rork Matriice
Muttimodal LLMs
Encoding
Palicy Head
Tepelogy
Token Embeeddnnge
Snadient Descomt
Spare mode
Enetvolking
Eniurdlng
Transble
Chass Aftemnian
Freezing
Munte Carfl Somple
Figure 3: Overview of PRAM. It consists of three core components: partition module to divide task into smaller sub-tasks, MLM-based agent module to to generate sub-task-specift answers, and adaptation module to efficiently learn global knowledge for MCF optimization.
NP-complete (Even et al., 1975) and finds many important applications across various domains (e.g.. communication, transportation, logistics and so on) as listed in Figure
PRAM: PARTITIONED RESOURCE ALLOCATION WITH MLMS
Complex MCF problems are often intractable as a whole but can be decomposed into subproblems defined over subsets of commodities and links. We propose to leverage MLMs for solving these subproblems in parallel, exploiting their mathematical reasoning capacity to yield high-quality allocations. We call this framework Partitioned Resource Allocation with MLMs (or PRAM for short). In the rest this section, we describe the motivations, benefits, and designs of PRAM.
3.1 MOTIVATIONS BEIHIND PRAM
Achilles' Heel L.--ased Methods The MCF problem typically formulated as a mathematical program, and in theory, optimal solution always be obtained (or infeasibility certified) using LPs. In practice, however, solving these programs can be prohibitively expensive: the worst-case complexity of LP with d variables is about (722.3729) (Lee & Sidford, 2015; Cohen et al., 2021; Narayanan et al., 2021), and modern systems may involve millions of variables (e.g.. at least one per source-destination pair), resulting in extremely long runtimes (e.g., in hours). Even worse, the optimal solution assumes full knowledge of all current commodities, yet real-time measurement scale is often impractical. As result, inputs are typically derived from historical flows their predictions. But given the uncertain and unpredictable nature of data such as network traffic (Wang al., 2006; Perry al., 2023), the final performance often highly suboptimal.
Where ML-based Methods Fall Short Recent studies have sought bypass the iterative optimization process using machine learning, where deep neural networks (DNNs) directly determine routing decisions. Although promising, existing MI.-based meth- ods fall short several key limitations, as we list in the follow- ing. ① High engineering costs. Their success heavily depen- dent on retraining, particularly for RL, which relies on fragile op- timization (Henderson al., 2018), and engineering DNN models, such as graph neural networks (GNNs) (Wu et al., 2020; Bernárdez al., 2021), for the target scenarios. which, however, can be labor- intensive due to the complex structures. ② Poor generulization. DNNs trained on specific environments may not perform well on unseen ones (Yuan et al., 2026). ④ Curse of dimensionality. Since representing the flow allocation needs O ([V)²) path weights, the scaling challenge remains. For instance, in a topology of 1,000 nodes with 4 candidate paths, the output alone would require 4 million dimensions.
Tradeaff Space
Progoeeuu
Hesritees
Tune Boyuund
PrAM
Generalization
(eg. RG)
Allocation Quality
Figure 2: Tradeoff space.
Faced with the above dilemma, an intuitive way of accelerating allocation and reducing complexity is to decompose MCF optimization into sub-tasks, applying powerful solver(s) simultaneously in each subproblem and merging their results at the end. For example, dividing commodities and demands evenly among A sub-problems will reduce the number of variables in each sub- problem by 122 (Cohen
3

---

Published as a conference paper at ICLR 2026
Outside
Target
Fietedelra Metriz
Flow
Source
Contact Puram
Profetvas
T'shon-bie E'ebeddagge
Baskore
Mi demand
MLM backbone
MEA backbone
MLM Beoldene
Key
Mk
Queryy
ong
Jobay Hesd
Volue
Adrattgge Palicy Gradhet
neght
Inside
(a) Inter-agent communication
(b) Reprogramming Iearned context embeddings
(c) MARL
Figure 4: Illustration of PRAM's adaptation framework. In (a), PRAM builds inter-agent communi- cation through LORA and reprogramming context using cross attention in (b). In (c), policy gradient flow is computed from each agent's 's difference reward to estimate the contribution of its actions to the team's 's global reward.
et al., 2021). Holding the premise, we argue that with adequate generalization and expressiveness, without frequent retraining, ML can also have the potential to be a principled alternative that enables near-optimal and rapid decision making for general MCF problems. This is exemplified by the recent popular pretrained (multimodal) large language models, opening new opportunities that allow PRAM to be effective, high-accuracy, but faster as seen in Figure 2.
3.2 MULTIMODAL PROBLEM PARTITION
To translate motivation into practice, the first step of PRAM is a partition procedure. We aim to maximize parallelism without excessive decomposition, since the latter can impede convergence and even increase solving time at scale. Revisiting the carlier example. decomposing the granularity of individual commodity flows may generate millions of subproblems, requiring numerous batching rounds even on modern GPUs So taking step back. PRAM focuses on solutions node level by treating commodities from the same source as one subset, reducing the model complexity from ((V))) to generally tractable ((V)). No need for any specially designed modules such as GNNs or RNNs, thanks to the off-the-shelf interface of MLMs, our design can be realized in a simple and convenient manner. As illustrated on the left of Figure 3, the procedure amounts to instructing the MLM with subgraphs (in image form) together with the commodity information (in text form). Concretely, we plot all routing links from the source node to every other node once to obtain a visual representation, which is then fed to MLMs through a vision encoder (e.g., CLIP (Radford et al., 2021)). At the same time, we introduce subtask-aware prompting, in which each recent demand in the commodity subset is explicitly paired with the subtask's 's key descriptions (e.g., source node specification) and statistics (e.g., historical rolling average demand). A prompt example is in Appendix C. The MLM "agent" then expected to jointly process and reason over both modalities to yield high-quality MCF configurations an issue we address next with our adaptation framework.
3.3 LIGHTWEIGHIT MULTI-AGENT ADAPTATION
In the implementation of PRAM, we promote its efficiency by sharing the MLM backbone among the subproblem agents. These logical agents can still behave differently because they receive different observations and thus evolve different hidden states. However, since the model is not inherently specialized for this task, the agents are unable to perceive each other's presence. This necessitates the introduction of additional (communication) parameters and raises the challenge of fine-tuning them.
Communication Our scheme of inter-agent communication is in Figure 4(a). The global (trainable) parameters are added in two ways: (i) Inside the model, we introduce low-rank matrices for MLM's attention weights to approximate the changes needed in the backbone parameters using the LoRA technique (Hu et al., 2022). (ii) Outside the model, inspired by LM's in-context learning (ICL. mechanics (Brown et al., 2020; Lin et al., 2024): an ability to flexibly adjust their prediction based on additional data given in context (i.e., in the input sequence itself), we create a set number of learnable "global context" embeddings as prefix of input prompts using the reprogramming technique (Li et al., 2023; Jin et al., 2024a). As illustrated in Figure 4(b), we connect the context to a frozen input embedding matrix of the tokenizer associated with the MLM, and we perform alignment with a multi- head cross attention layer. Specifically, we model text prototypes as keys and values by linearly aggregating original token embeddings, while the context parameters act queries to extract
4

---

Published as a conference paper at ICLR 2026
Conmulity
Figure 5: A case study on the multi-commodity flow problem. (left) In a simple network, both ! and 2 need to transmit commodities to Q each having two candidate paths. (right) The (expected) objective functions are plotted against the path weights, with their optima indicated by red dots. The curves exhibit clear convexitylconcavity, lending theoretical support to the soundness of PRAM.
global information or guidance the agent requires and understands. Upon packing the prompt and context embeddings, as shown in Figure 3, we feed them through the MLM backbone to produce the agent-decision emheddings. As our evaluation results in $ 5.4 show, these designs impose limited parameter overhead; in other words, PRAM is lightweight overall.
Adaptation While independent adaptation of different agents via gradients from the global objective (loss) is straightforward, the lack of information exchange during fine-tuning limits the ability to learn coordinated strategies to assess individual agent's 's contribution to the system. To overcome this, we adopt multi-agent reinforcement learning (MARL) algorithms (Busoniu et al., 2008; Kraemer & Banerjee, 2016; Foerster et al., 2018; Lyu al., 2021; 2023). In particular, PRAM leverages counterfuctual reasoning (Foerster et al., 2018; Xu et al., 2023), asking: how would the global objective change if only one subser's 's flows were reallocated while others remained fixed? The reward difference defines the single agent's 's advantage relative to a counterfactual baseline, thereby quantifying its contribution the joint outcome.
To be more specific, when (history) demands arrive, PRAM provides the token emheddings of each subset as the local state s; to the corresponding RL agent which then outputs an action a; from a policy head network, i.c., vector of path weights for its managed demands. All agents share a policy wo parameterized by 6, trained via policy gradient. We then denote by s the central state composed of all local states $;, and by a the joint action of all agents' actions a;. After all decisions are made, a reward R(s a) is obtained.
compute the advantage A, (s. a), PRAM exploits the one-step nature of the MCF; since actions (flow allocations) do not affect future states (demands or/and paths), the expected return reduces to the immediate reward R(s, a). Morcover, if agent i changes its action to af while others keep theirs fixed, the new joint action (a_i, d can be directly evaluated by simulating its effect on the objective, i.c., we compute the MCF objective obtained if the new joint action were to be used. Putting it together, PRAM computes the advantage for agent
as(s.a = -((s - ETalddss)RRsss
g = Ex
where g is the policy gradient, and the counterfactual baseline is approximated via Monte Carlo sampling, i.e., drawing random actions af ~ [=(-ss:) As shown in Figure 4(c), we fine-tune PRAM in an end-to-end manner, with 8 denoting all tunable parameters, so that gradients are backpropagated seamlessly from the policy head network. We include more details of the algorithm in Appendix C.3.
UNDERSTANDING PRAM: CASE STUDY AND THEORY
Through the Looking Glass MCF To delve into the MCF problem, we conduct an illustrative case study in Figure 5. This study is based on a toy topology consisting of 4 nodes and 5 links,
each with a capacity of 1. Now, suppose nodes ① and ② need to send commodity flows to @
With equal probability, their demands is either 6 or 7
R The candidate paths from ① to
are (1, and (1, 3.3.44 4), while those from node 2 to + are (2,4) and (2,3, 4). In this case, the configuration x is fully determined once the weights on links (1,4) and (2, 4) are specified. In Figure 5 (right), we plot the curves of all three (expected) objective functions as the two weights vary, and mark the optimal values with red dots. We observe that the objective possesses a favorable
5

---

Published as a conference paper at ICLR 2026
property- wot.e//t/oooooccoiiitttyyy with respect to the path weights. such property ensures that simple first-order methods, i.c., gradient descent (GD), can effectively solve the problem. In particular, by iteratively updating the configuration in the direction of the steepest descent, one can guarantee convergence to the global optimum. We formalize this insight in the following theorem:
Theorem 1. (Solving MCF with GD) Consider GD algorithm with update rule ((++1) = x 178--
Then, there exists a step size dl > O and a finite number of iterations T such that ((x), the MCF objective function, attains the optimum up an arbitrarily small error.
Appendix D.3 presents the proof of Theorem 1, which hinges on the convexity/coneavity of the three objective functions. In our simple example, executing GD) requires precise knowledge of the distribution of demands. However, It is almost impossible to estimate the expected objective without exact prediction of the future demands. To address this, a more practical approach is to implicitly model the probability distribution using a DNN trained on extensive empirical data (like PRAM).
What Makes PRAM Tick Given the above case study, to approximate the optimal mapping from recent observations to configurations, the theoretical analysis of PRAM'S effectiveness is intrinsically aligned with the GD algorithm. Before proceeding, we first present the convergence guarantee of PRAM'S multi-agent adaptation:
Lemma 1. Assume that both R(s,a) and its Hessian ViR(s, a) are bounded. Let { be any step-size sequence satisfying lim 04k = 0, Ex 0Kk = oo; and ((k+1) = g(k) + Oy.Uk, where uk in the direction of the gradient. Then the policy ileration converges such that the expected gradient lims-soo Eg
>>; Ai(s, a) VocEs log To0o (a; 8:)] = O.
The proof Lemma 1 (see Appendix D.4) refers the approximation results from previous actor- critic algorithms (Bertsekas & Tsitsiklis, 1996; Sutton et al., 1999). Lemma 1 establishes the convergence of PRAM a locally optimal agent, which implies that, once the fine-tuning complete, our model can be regarded as global mapping from observed demands to allocation schemes. Now back MCF problems, our intuition that MLMs are capable of running the GD algorithm again in the light of ICL. Concretely, it can approximate gradient-based few-shot learning within its forward pass (Fu et al., 2023), thereby "reasoning" its way to the solution. We formalize this intuition in assume the MCF problem in the token space of PRAM as linear regression. Theorem 2. (PRAM Learns to Implement GD) Let the learned context and the problem be well- defined under Assumption 2 and 3 in the Appendix. Then there exists an adapted MLM with constant depth and constant width that can simulate multiple steps of GD updates on the problem objective.
Theorem 2, where
The proof deferred to Appendix D.5, which follows a simple weight construction introduced by Ahn et al. (2023). Taken together, the convexity of MCF problems (Theorem 1) and the ability of MLMs to simulate gradient descent (Theorem 2) establish that PRAM, once adapted, internally acts as an optimizer on the distance between initial configurations (input token embeddings) and near-optimal -oonsssttt with our empirical results (S 5).
configurations fagent-decision
5
MAIN RESULTS
We evaluate PRAM on open- multimodal language models, with Instruct (Bai et al., 2025) as a standard. Unless otherwise noted, we truncate and fine-tune the first& 8 layers of its LM as the backbone. Implementation specifics are in Appendix C. and key experi- mental setups are summarized below with further details Appendix B. This section sequentially includes comparative experiments real-world and large-scale generated datasets, generalization analysis, and ablation studics. We also provide additional supplementary experiments (e.g.. w.r.t number of layers, used MLMs, and demand distributions) in the Appendix E.
VL-
Datasets. Our evaluation is conducted on two groups of datasets. First, we leverage five real-world datasets, namely Meta DR, Meta WEB, Abilene, CERNET, and GÉANT. These datasets are relatively small in scale, each containing fewer than 30 nodes. To complement it, also incorporate five large-scale topologies with synthetic data, including GtsCe, Colt, UsCarrier, Cogentco, and Kdl. The sizes of these topologies range from 100 to 800 nodes. In $ 5.2, we employ gravity model (Roughan et al., 2002) to generate synthetic demands. Additionally, we select the /2 most recently observed demands as historical information.
Baselines. We compare PRAM with six representative baselines: linear programming (LP (Gurobi Optimization, LLC,, 2024)), partition-based optimization (POP (Cohen al., 2021)), heuristic

---

Published as a conference paper at ICLR 2026
2.00
Max: Link Usagee (4)
1.50
0.00
1.00
2.00
Tosl Floek
840
Avg. Canair Plow (1)
1.00
1.00
Mara-00 08
Abllnn
CERNET
GEANT
DRL
HA/RP
Jaredi
POP (ared)
LP-t0 (pric)
Figure 6: Real-world evaluation results of maximum link utilization (lower the better), allocated total flow (higher the better), and concurrent flow (higher the better). PRAM achieves top-tier performance.
approach (LP-top (Namyar al., 2022)), reinforcement learning (DRL (Valadarsky et al., 2017)), and GNN-based machine learning (HARP (AIQiam et al., 2024), Aether (Fan al., 2025)). Since some methods rely the ground- truth demand, also evaluate their predicted variants (pred) using the default moving-avenna
forecasting, which we found to be simple yet effective (see Appendix E.3).
Metrics. We consider three standard objectives for the MCF problem: minimizing maximum link utilization (MLU), maximizing total flow, and maximizing concurrent flow. To enable consistent comparison across datasets, all reported results normalized, which is the ratio of the result the (near-)optimal result obtained using the LP optimization solver. We also measure the computation time of each approach on the same machine assess their efficiency.
5.1 EXPERIMENT ON WEALDWORLD DATASETS
Our first experiment aims to answer: how does PRAM perform in practice? We present the results of the experiment in Figure 6, which evaluates the performance of our proposed PRAM over publicly available real-world datasets. In total, run three independent sub-experiments, one for each MCF allocation objective. First, the results reveal substantial room for improvement when flows allocated by the prior ML-based approach (i.e., DRL), likely due the well-known training challenges of RL with randomly initialized networks. Second, the gap between the optimal solution and the partitioned solution produced by POP evident, making POP the second-worst method. Third, while LP-top often performs comparably to LP, it exhibits less stability because it focuses exclusively on optimizing large flows. From the figure, we can also see that relative to their exact- demand counterparts, all predictive variants suffer performance degradation (no less than 10%). with the note that the MI.-based methods use historical demand as input. Finally, we find PRAM achieves the second-best average performance across the three objectives, ranking just behind the L.P solver that has perfect future prediction, indicating it is able to achieve ncar-optimal by leveraging the reasoning power of MLMs. One surprising observation is that, in the case of minimizing link usage, PRAM'S 's output configurations are even better than those of LPs (in particular, about 21 % lower in CERNET and 45% in GÉANT). This may be because MLU exhibits stronger convexity cecerties--onssstter with our theoretical results and demonstrates more stable variation ranges, making it especially amenable to fine-tuning with MLMs. HARP performs second only to ours in MLU because it is specifically optimized for this objective. However, its performance the other two objectives is subpar. In a clear example, on the CERNET and GÉANT datasets, its average allocated concurrent flow is even inferior to the simple DRL algorithm. While Aether shows an improvement over DRL, it fails to overcome the inherent RL limitations, resulting in overall trend that is still inferior to PRAM and classical method. In conclusion, these findings suggest PRAM's good suitability for real-world deployment.
7

---

Published as a conference paper at ICLR 2026
Normalized Results w.r.t Computation Time (s)
Ukagee
140
130
1.00
120
110
1.0
1.0
1.0
100
120
Toal
8:05-
135
0
0.80
119
6.0 18
0.10
Avg. Contur Flow(1) (1)
8.08
ESC
0.0
10
Oogantso
HAMP
Arter
PC
POP
LPo0p
LPAk greef
Figure 7: Comparison of PRAM with baselines on large-scale topologies using synthetic data. As the solution space expands, PRAM demonstrates scalable performance, achieving comparable or superior results with reduced computation time.
5.2 EXPERIMENT ON LARGE-SCALLE DATASETS
Next, we experiment on larger network topologies to explore the following question: how does PRAM work at scale? Analogously. Figure 7 compares online testing results of different methods together with their running time. As the network size grows. PRAM demonstrates the intended scalability due to dividing operation, delivering high-quality allocations with lower computation time. In particular, on the largest Kdl topology (754 nodes, 1,790 links, and over million candidate path weights), PRAM completes each flow allocation in under 25 seconds on average- 55 faster than POP, 7x faster than LP-top, and 100x faster than LP. For demand prediction, our non-parametric approach introduces negligible overhead. DRL, benefiting from small policy network without MLM's backbone, achieves the fastest inference, but fails to approach near-optimal allocation quality. This is because the simple network also limits its expressive capability and lacks effective modeling of complex structures. In contrast, PRAM satisfies > 90% of the performance of the best-performing LP scheme that close enough to the optimal. Additionally, we outperform HARP and Aether on average by 6.1% 1% and 17.2% MLU, by 16.6% and 7.3% on total flow, and by 24.8% and 13.5% on concurrent flow. HARP's recurrent optimization and invariant architecture limits its expressive capability, while Aether's link-path graph lacks effective modeling of large-scale topology structures and correlations. Since we inject Gaussian noise to mimic temporal demand variations, prediction- based schemes achieve slightly better results than with real-world traces, yet still lag behind PRAM. In summary. ML-based methods are fast but inaccurate, while LP-based methods perform well but are inefficient. By striking a balance, PRAM offers both efficiency and accuracy, making it compelling choice for production MCF systems.
5.3 GENERALIZATION TO UNSEEN SITUATIONS
In this subsection, we assess our model's performance on unforeseen conditions, i.e., asking: can PRAM generalize well? Due to space constraints, we hereafter report results primarily in terms of link utilization. We consider two most representative situations, where unforeseen changes affect the two inputs of the MCF problem: topology (link failure) and commodity demand (flow fluctuation).
(1) Coping with link failures. PRAM addresses link failures by first reassigning the affected com- modity flows using simple heuristics. Take the case of three candidate paths having weights of (2. 3: ). If the first path experiences failure, the adjusted weights would be proportionally reset as (0. 5 2). Such reassignment requires minimal computation time, and the entire process operates at the millisecond-level scale, comparable to a standard inference step. In addition, the capacity of failed links will be marked as zero, and will be depicted in the sub-image, which is then fed into
PRAM. Figure 8(a) compares the performance of PRAM with prediction-basse methods (access to
8

---

Published as a conference paper at ICLR 2026
HAP
Li- gind
0-
Porfrrance Degratation Ph)
10
-15-
-20
Nao 02
Pun
22-
DRL
HARP
Aahee
-95
PCP 1pret)
LP gred
LP-ko (aied)
525
D.5
1.0
20
Number Link Falurea
Pluduation Pactor
(a) MLU change with random link failures
(b) Performance decline with fluctuation
Figure 8: Handling different unforeseen conditions on GÉANT; The right plot shows the mean and
deviation from 10th to 90th percentile. The performance variations of PRAM are not significant.
Average (normalized) Maximum Link Utilization (
Number of TTainabee Paramccors (460)
30008
Compubtter rand!
3000
We man
2000
wis lors
15008
Nis cmn
1009
minn
0000
Pran
OSANT
aack
1152
Abllene
25
Kdl
(a) # of trainable parameters with(out) partition
(b) Ablation study of PRAM's key components
Figure 9: Scalability and link utilization performance comparison between PRAM and its variants.
failures), in terms of link utilization, when different numbers of links fail in the GÉANT topology. For a fair comparison, we applied the same correction on all ML-based baselines. It can seen that, except for HARP, they all have showed significant decline, with approximately 7% to 24% more MLU upon increased link failures. Our results show that PRAM still consistently outperforms all baselines without re adaptation, indicating its generality across (transient) failures.
(2) Reacting to flow fluctuation. Our approach for increasing flow variability as follows: we intro- duce independent noise into each destieddstination pair by sampling from Gaussian distribution N(H, o² d² where μ and as.d is the standard deviation of the demand Dz,a- These noises are then multiplied by factor a control the fluctuation intensity. Also, figure 8(b) shows that the per- formance degradation of PRAM is no higher than 15% even for a = 2 in the 90th percentile. Almost all the evaluated ML-based baselines handle small fluctuations effectively, but their performance declines noticeably as the fluctuations escalate to a > 0.5. Meanwhile, the three classical methods struggle in this scenario, with a drop more than twice that of PRAM.
5.4 ABLATION STUDY AND VISUALIZATION
At the end of this section, we would like to investigate the final question: how is PRAM effective? We perform an ablation study to assess the impact of PRAM'S 's key features on its overall performance.
(I) Impact of Problem Partition. Figure (a) compares the number of tunable parameters in PRAM with its non- aon-paaitiooned variant, i.e., PRAM w/o partition (or w/o par. in short), across topologies of different sizes. Without partitioning, the parameter scale grows uncontrollably; for example, on the KDL. topology, the number of trainable parameters is about 31,600 MB, which nearly matches that of full-parameter MIM adaptation. In contrast, PRAM localizes parameters within LoRA and context modules- both inherently lightweight as shown in the figure-so its model size hardly increases with network scale. These results underscore the critical role of problem partition in handling MCF optimizations with large solution spaces.
(2) Effect of Adaptation Choices. To assess the importance of PRAM'S 's core components, we retrain five variants the Abilene and KDL topologies: ! PRAM w/o mlm: the MLM backbone is removed, with tob-topologies handled by a GNN and demands by FC layers; 2 PRAM w/o con.: the global context excluded from the input prompt; 3 PRAM w/o lora: all low-rank adapters are removed from

---

Published as a conference paper at ICLR 2026
Grosa Atlention Mep
Lawr (Contee)) Enkoeding
Agprep Popeppee Teat Proteppa
Word Set
Word Sert
Word Set
(a) Cross-attention map
(b) Visualization of ten different learned text prototypes
Figure 11: Learned representation interpretation of the context embeddings. In (a), each row represents a context embedding, while columns correspond to selected words. In (b), each row represents a linear aggregated prototype, while columns correspond words from different sets.
the backbone; A PRAM w/o marl: the model is directly fine-tuned with the objective function end to end; ( PRAM w/o par.: identical PRAM w/o partition. The results are presented in Figure (b).
We can see that using MLM significantly improves performance in minimizing MLU, especially for Abilene. Moreover, all three designs (i.e., LoRA, learned context and multi-agent RL) used in our adaptation contribute to the optimization of the ob-
From From
Maximum Link Unlitation (4)
jective function. Although PRAM w/o partition achieves a lower MLU on the smaller network, it is
too "gigantic" to make decisions on large Kdl. As
generality test, moreover, we evaluate the distinct single-faiture scenarios and and artificially inject the 24 most influential failures to the GÉANT network. Figure 10 compares the performance of PRAM with its variant excluding MLM. Across all failure sce-
Figure 10: (un-normalized) MI. under differ- ent single-link failure scenarios for GÉANT. narios, PRAM'S median MLU ranges from 1.4 to 2.3, while that of PRAM w/o MLM ranges from 1.5 to 3 which is clearly not as effective under link failures because rescaling is not as effective as recomputing routes. confirms PRAM'S 's superiority to handle topology change not observed in the training data, by leveraging the pre-trained knowledge of MLMs.
(3) Visualization of Learned Representation. We present an interpretation of the learned context module on the Abilene dataset in Figure 11. As shown in Figure 1 l1 we visualize the cross-attention map of a single head by replacing text prototypes with token embeddings of words either related or unrelated to the task. Each cell reflects the relevance between a row and a column (with brighter values indicating stronger correlation).
further examine how context is represented through different prototype combinations, we randomly select ten prototypes and visualize them in Figure 1 (b)... For this analysis, we consider three word sets - word set (commodity): "FFlow", "Demand"... word set 2 (topology): ("Capacity", "Node"... .... J. and word set 3 (others): { ("Monday", "February". ... ]. From the heatmaps, we make two key observations: first, prototypes exhibit strong relevance to words describing MCF problems; and second, PRAM successfully aligns its learned context with task-relevant word embeddings. So we can reasonably conclude that context satisfies our assumption.
CONCLUSION
In this paper, we present PRAM, a novel MCF solution that takes advantage of MLMS to optimize practical flow allocation performance. The MLM is equipped with a partition method to "divide" complexity and transform the original problem into a multi-agent decision-making problem. Then we design an adaptation (and communication) framework to "harmonize" the learning of these agents. We also theoretically study the effectiveness of PRAM in well-studied theoretical models. Our evaluations demonstrate that the PRAM is accurate, fast and robust, highlighting the potential of leveraging MLMs for "shooting" MCF problems. However, fine-tuning PRAM remains resource- intensive, even after truncating the backbone model. In addition, we acknowledge that the visual encoding scheme may introduce inevitable bias, discussed in Appendix E.5 & E.6. Due space constraints, we defer discussion of several important limitations regarding PRAM and potential future directions to Appendix F.
10

---

Published as a conference paper at ICLR 2026
7 ETHICS STATEMENT
We did not use any non-public data, unauthorized software, or API in our paper. In particular, the measured demand matrices, used in evaluation, aggregate counters between pairs of the granularity of minutes (or coarser). They do not contain user IP addresses or packet contents. Thus, there are no privacy or other related ethical concerns.
REPRODUCIBILITY STATEMENT
The paper fully discloses all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and conclusions of the paper. Specifically, we detailed all the experimental setups and implementation details in 5 and Appendix B & C. Moreover, experimental codebase is publicly available at https: / thub. com/ www.////o//ooeeeeeeeeeeeeeeee Y-debug-sys/PRaM
ACKNOWLEDGMENT
This work is supported in part by the Key Research and Development Program of Zhejiang Province under Grant No. 2025C02103, in part by the National Natural Science Foundation of China under Grant No. 92373205 and No. 62572168, in part by the National Key Research and Development Program of China No. 2023YFB4404401, and in part by the Anhui Provincial Natural Science Fund (2508085MF151).
REFERENCES
Firas Abuzaid, Srikanth Kandula, Behnaz Arzani, Ishai Menache, Matei Zaharia, and Peter Bailis. Contracting wide-area network topologies solve flow problems quickly. In 18th USENIX Symposium on Networked Systems Design and Implementation (NSDI 21), pp. 175-200, 2021.
Cited on pages 2 and 20.
Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report. arXiv preprint ar 2r33v:2303.088 087774, 2023. Cited page 20.
Kwangjun Ahn, Xiang Cheng, Hadi Daneshmand, and Suvrit Sra. Transformers learn implement preconditioned gradient descent for in-context learning. Advances in Neural Information Processing Systems, 36:45614 45650, 2023. Cited pages 6, 21, 33, 34, and 42.
Nasrin Akhter and Mohamed Othman. Energy aware resource allocation of cloud data center: review and open issues. Cluster computing, 1933::163-1118) 2016. Cited on page 20.
Ekin Akyürek, Dale Schuurmans, Jacob Andreas, Tengyu Ma, and Denny Zhou. What learning algorithm is in-context learning? investigations with linear models. In The Eleventh International Conference on Learning Representations, 2023. Cited on page 21.
Abd AIRhman AlQiam, Yuanjun Yao, Zhaodong Wang, Satyajeet Singh Abuja, Ying Zhang, Sanjay G Rao, Bruno Ribeiro, and Mohit Tawarmalani. Transferable neural wan te for changing topologies. In Proceedings of the ACM SIGCOMM 2024 Conference, pp. 102, 2024. URL https: ///ithub.cc. com
htt.:/cco/uuuueuee Pwwpue- Macieee-SLHHAAA Cited on pages 7, 25, and 32.
David Applegate and Edith Cohen. Making intra-domain routing robust to changing and uncertain traffic demands: Understanding fundamental tradeoffs. In Proceedings of the 2003 conference Applications, technologies, architectures, and protocols for computer communications, pp.
313-324, 2003. Cited on pages 1, 20, and 24.
Arjang A Assad. Multicommodity network flows-a survey. Networks, 8(1):37-91, 1978. Cited on page 1.
Yossi Azar, Edith Cohen, Amos Fiat, Haim Kaplan, and Harald Racke. Optimal oblivious routing in polynomial time. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing. pp. 383-388, 2003. Cited on page 20.
11

---

Published as a conference paper at ICLR 2026
Shuai Bai, Keqin Chen, Xuejing Liu, Jialin Wang, Wenbin Ge, Sibo Song, Kai Dang, Peng Wang, Shijie Wang, Jun Tang, et al. Qwen2. 5-vl technical report. arXiv preprint arXiv:2502.13923, 2025. URL https: htt.ghhggggngccegeeeeeeeeceeaoae co/ weweeiiinn//On Cited on pages and 43.
Burcu Balcik, Scyed Iravani, and Karen Smilowitz. Multi-vehicle sequential resource allocation for a nonprofit distribution system. IE Transactions, 46(11):1279 1297, 2014. Cited on pages 1 and 20.
Guillermo Bernárdez, José Suárez-Varela, Albert López, Bo Wu, Shihan Xiao, Xiangle Cheng, Pere Barlet-Ros, and Albert Cabellos- Cabellos-Aparicio Aparicio. Is machine learning ready for traffic engineering optimization? In 2021 IEEE 29th International Conference Network Protocols (ICNP), pp. 1-11. IEEE, 2021. Cited on pages 1, 3, and 20.
Dimitri Bertsekas and John Tsissiklis. Neuro-Dynamic Programning. Athena Scientific, 1996. Cited on pages and 33.
Niels Blaauwbroek, Phuong H Nguyen, Mente J Konsman, Huaizhou Shi, Ren IG Kamphuis, and Wil L. Kling. Decentralized resource allocation and load scheduling for multicommodity smart energy systems. JEEE Transactions on Sustainable Energy, 154):1500- 2015, Cited on pages and 20.
Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances neural information processing systems, 33:1877-1901, 2020. Cited page 4.
Lucian Busoniu, Robert Babuska, and Bart De Schutter. A comprehensive survey of multiagent reinforcement learning. JEEE Transactions on Systems, Man, and Cyhernetics, Part C (Applications and Reviews),
2008. Cited on page 5.
Konstantinos Chalvatzis, Hanif Malekpoor, Nishikant Mishra, Fiona Lettice, and Sonal Choudhary. Sustainable resource allocation for power generation: The role of big data enabling interindustry architectural innovation. Technological Forecasting and Social Change, 144:381-395 2019. Cited on page 20.
Ching Chang, Wen- Chih Peng. and Tien-Fu Chen. Llm4ts: Two-stage fine-tuning for time-series forecasting with pre-trained lmms. CoRR, 2023. Cited on page 21.
Fangzhe Chang, Jennifer Ren, and Ramesh Viswanathan. Optimal resource allocation in clouds. In 2010 IEEE 3rd International Conference on Cloud Computing, pp. 418-425. IEEE, 2010. Cited on pages and 20.
Li Chen and Mingquan Ye. High-accuracy multicommodity flows via iterative refinement. In 5 International Colloquium on Automata, Languages, and Programming (ICALP 2024), pp. 45-1. Schloss Dagstuhl Dagstuhl-Leibnizz Zeitriz-Zentrum für Informatik, 2024. Cited on page 1.
Runjin Chen, Tong Zhao, Ajay Jaiswal, Neil Shah, and Zhangyang Wang. Llaga: large language and graph assistant. In Proceedings of the 41st International Conference on Machine Learning, pp. 7809-7823, 2024. Cited on page 21.
Xingwu Chen and Difan Zou. What can transformer learn with varying depth? case studies on sequence learning tasks, In International Conference on Machine Learning, pp. 7972-8001. PMLR. 2024. Cited on page 21.
Michael Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the current matrix multiplication time. Journal of the ACM (JACM), 68(1):1-39, 2021. Cited on pages 1, 2, 3, and 6.
Shimon Even, Alon Itai, and Adi Shamir. On the complexity of time table and multi-commodity flow problems. 16th annual symposium on foundations of computer science (sfcs 1975), pp. 184-193. IEEE, 1975. Cited on page 3.
12

---

Published as a conference paper at ICLR 2026
Yu Fan, Jingyao Liu, Pengjin Xie, and Liang Liu. Aether: Toward generalized traffic engineering with elastic multi-agent graph transformers. In JEEE INFOCOM 2025-IEEE Conference on Com- puter Communications, pp. 1-10. IEEE, 2025. URL https: /github com/ fanyubupt/ Aether. Cited on pages 7 and 26.
Bahare Fatemi, Jonathan Halcrow, and Bryan Perozzi. Talk like graph: Encoding graphs for large language models. In The Twelfth International Conference on Learning Representations, 2024. Cited on pages 21 and 47.
Jakob Foerster, Gregory Farquhar, Triantafyllos Afouras, Nantas Nardelli, and Shimon Whiteson. Counterfactual multi-agent policy gradients. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018. Cited on pages 2, 5, 33, and 40.
Deqing Fu, CHEN Tianqi, Robin Jia, and Vatsal Sharan. Transformers learn higher-order optimization methods for in-context learning: A study with linear models. 2023. Cited on page 6.
Javier González and Aditya Nori. Does reasoning emerge? examining the probabilities of causation in large language models. Advances in Neural Information Processing Systems, 37:1 17:1177771117711 2024. Cited page 21.
Gurobi Optimization, LLC. Gurobi Optimizer Reference Manual, 2024. URL https: //www. gurobi com. Cited pages 6 and 26.
Aric Hagberg. Pieter Swart, and Daniel A Schult. Exploring network structure, dynamics, and function using networkx. Technical report, Los Alamos National Laboratory (LANL), Los Alamos, NM (United States), 2008. Cited on page 28.
Elad Hazan, Kfir Levy, and Shai Shalev-Shwartz. Beyond convexity: Stochastic quasi-convex optimization. Advances in neural information processing systems, 28, 2015. Cited on page 34.
Peter Henderson, Riashat Islam, Philip Bachman, Joelle Pincau. Doina Precup, and David Meger. Deep reinforcement learning that matters. In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018. Cited page 3.
Oliver Hope and Eiko Yoneki. Gddr: Gun-based data-driven routing. In 2021 IEEE 41st International Conference Distributed Computing Systems (ICDCS), pp. 517-527. IEEE, 2021. Cited on page 20.
Edward J Hu, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, Weizhu Chen, et al. Lora: Low-rank adaptation of large language models. In The Tenth International Conference Learning Representations, 2022. Cited page 4.
Jianhao Huang, Zixuan Wang, and Jason D Lee. Transformers learn to implement multi-step gradient descent with chain of thought. In The Thirteenth International Conference on Learning Representations, 2025. Cited on page 21.
John D Hunter. Matplotlib: A 2d graphics environment. Computing in science & engineering, 9(03): 90-95, 2007. Cited on page 28.
Sushant Jain, Alok Kumar, Subhasree Mandal, Joon Ong, Leon Poutievski, Arjun Singh, Subbaiah Venkata, Jim Wanderer, Junlan Zhou, Min Zhu, et al. B4: Experience with a globally-deployed software defined wan. ACM SIGCOMM Computer Communication Review, 43(4):3-144 14, 2013. Cited on page 20.
Ming Jin, Shiyu Wang, Lintao Ma, Zhixuan Chu, James Y Zhang, Xiaoming Shi, Pin- Yu Chen, Yux- uan Liang, Yuan-Fang Li, Shirui Pan, et al. Time-llm: Time series forecasting by reprogramming large language models. In The Twelfth International Conference Learning Representations, 2024a. Cited on pages 4 and 21.
Ming Jin, Yifan Zhang, Wei Chen, Kexin Zhang, Yuxuan Liang, Bin Yang, Jindong Wang, Shirui Pan, and Qingsong Wen. Position: What can large language models tell us about time series analysis. In Forty-first International Conference Machine Learning, 2024b. Cited on page 21.
13

---

Published as a conference paper at ICLR 2026
Narendra Karmarkar. A new polynomial time algorithm for linear programming. In Proceedings of the sixteenth anmual ACM symposium on Theory of computing, pp. 302-311, 1984. Cited on page 1.
Leonid G Khachiyan. Polynomial algorithms in linear programming. USSR Computational Mathe- matics and Mathematical Physics, 20(1):53-72, 1980. Cited on page 1.
Diederik Kingma. Adam: A method for stochastic optimization. arXiv preprint arXiv: 1412.6980,
2014. Cited on page 32.
Simon Knight, Hung X Nguyen, Nickolas Falkner, Rhys Bowden, and Matthew Roughan. The internet topology zoo. IEEE Jourmal on Selected Areas in Communications, 29(9):1755-1775.
2011. URL http: / /wwww topology- z00. org/ Cited on page 23.
Igor V Konnov. On convergence properties of subgradient method. Optimization Methods and Software, 18(1):53-66) 62, 2003. Cited on page 34.
Landon Kraemer and Bikramjii Banerjee. Multi-agent reinforcement learning as a rehearsal for decentralized planning. Neurocomputing, 190:82-94, 2016. Cited on page
Umesh Krishnaswamy, Rachee Singh, Nikolaj Bjørner, and Himanshu Raj. Decentralized cloud wide-area network traffic engineering with (BLASTSHIELD), In 19th USENIX Symposium Networked Systems Design and Implementation (NSDI 22), pp. 325 338, 2022. Cited on page 20.
Praveen Kumar, Yang Yuan, Chris Yu, Nate Foster, Robert Kleinberg, Petr Lapukhov, Chiun Lin Lim, and Robert Soulé. (Semi-oblivious) traffic engineering: The road not taken. In 15th USENIX Symposium on Networked Systems Design and Implementation (NSDI 18), pp. 157-170, 2018. Cited on page 20.
Yin Tat Lee and Aaron Sidford. Efficient inverse maintenance and faster algorithms for linear programming. In 2015 IEEE 56th annual symposium on foundations of computer science, pp. 130-249 IEEE, 2015. Cited on page 3.
Junnan Li. Dongxu Li, Silvio Savarese, and Steven Hoi. Blip-2: bootstrapping language-image pre-training with frozen image encoders and large language models. In Proceedings of the 40th International Conference on Machine Learning, 2023. Cited on page
Xing Li, Congxiao Bao, Maoke Chen, Hong Zhang, and Jianping Wu. The china education and research network (cernet) ivi translation design and deployment for the ipv4/ipv6 coexistence and transition. Technical report, 2011. Cited on page 23.
Zhiyuan Li, Hong Liu, Denny Zhou, and Tengyu Ma. Chain of thought empowers transform- to solve inherently serial problems. In The Twelfth International Conference Learning Representations, 2024. Cited on page 21.
Licong Lin, Yu Bai, and Song Mei. Transformers as decision makers: Provable in-context reinforce- ment learning via supervised pretraining. In The Twelfth International Conference on Learning Representations, 2024. Cited on pages 4 and 21.
Chenyi Liu, Haotian Deng, Vaneet Aggarwal, Yuan Yang, and Mingwei Xu. Shooting large-scale traffic engineering by combining deep learning and optimization approach. Proceedings of the ACM on Networking. 3(CoNEXT1):112) 2025. Cited on page 32.
Yong Liu, Guo Qin, Xiangdong Huang, Jianmin Wang, and Mingsheng Long, Autotimes: Autoregres- sive time series forecasters via large language models. Advances in Neural Information Processing Systems,
2024. Cited on page 51.
Zhiyuan Liu, Sihang Li, Yanchen Luo, Hao Fei, Yixin Cao, Kenji Kawaguchi, Xiang Wang, and Tat-Seng Chua. Molca: Molecular graph-language modeling with tross-modal projector and uni-modal adapter. In EMNLP, 2023. Cited page 21.
Xueguang Lyu, Yuchen Xiao, Brett Daley, and Christopher Amato. Contrasting centralized and decentralized critics in multi-agent reinforcement learning. In Proceedings of the 20th International Conference Autonomous Agents and MultiAgent Systems, pp. 844 852, 2021. Cited on page 5.
14

---

Published as a conference paper at ICLR 2026
Xueguang Lyu, Andrea Baisero, Yuchen Xiao, Brett Daley, and Christopher Amato. On centralized critics in multi-agent reinforcement learning. Journal of Artificial Intelligence Research, 77: 295-354, 2023. Cited
page 5.
Arvind V Mahankali, Tatsunori Hashimoto, and Tengyu Ma. One step of gradient descent provably the optimal in in-context learner with one layer of linear self attention In The Twelfth International Conference on Learning Representations, 2024. Cited on page 21.
Marvin B Mandell. Modelling effectiveness- equity trade-offs public service delivery systems. Management Science, 37(4)) 1991. Cited on page 20.
Meta.
2024.
URL "https:/ /hopg::://hggiiiii www.ggginnaacccooom co/ Cited on page 43.
meta- meeta-llamal 1aee--1iaaaaaaimm---------------
Pooria Namyar, Behnaz Arzani, Ryan Beckett, Santiago Segarra, Himanshu Raj, and Srikanth Kandula. Minding the gap between fast heuristics and their optimal counterparts. In Proceedings of the 21 ACM Workshop on Hot Topics in Networks, pp. 148, 2022. Cited on pages 7 and 26.
Deepak Narayanan, Fiodar Kazhamiaka, Firas Abuzaid, Peter Kraft, Akshay Agrawal, Srikanth Kandula, Stephen Boyd, and Matei Zaharia. Solving large-scale granular resource allocation problems efficiently with pop. In Proceedings of the ACM SIGOPS 28th Symposium on Op- erating Systems Principles (SOSP 21), pp. 521-537, 2021. URL https: / / gight........ com/ stanford- fattluuturadaaaaaa /POP. Cited pages 3, 20, and 26.
Eshaan Nichani, Alex Damian, and Jason D Lee. How transformers learn causal structure with gradient descent. In Proceedings of the 41st International Conference on Machine Learning, pp. 03018-38070, 2024. Cited on page 21.
Md Noor-A-Rahine Zilong Liu, Haeyoung Lee, GG Md Nawaz Ali, Dirk Pesch, and Pei Xiao. survey resource allocation in vehicular networks. IEEE transactions intelligent transportation systems, 23(2):001-721, 2020. Cited on page 20.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, birh-ppeformance deep learning library. Advances in neural information processing systems, 32, 2019. Cited on page 27.
Yarin Perry, Felipe Vieira Frujeri, Chaim Hoch, Srikanth Kandula, Ishai Menache, Michael Schapira, and Aviv Tamar. (DOTE): Rethinking (predictive) (WANN traffic engineering. In 20th USENIX Symposium on Networked Systems Design and Implementation (NSDI 23), pp. 1557-1581, 2023.
Cited on pages 3 and 33.
Harald Racke. Minimizing congestion in general networks. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pp. 43-52, IERE, 2002. Cited on page 20.
Alec Radford, Jong Wook Kim, Chris Hallacy. Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International conference on machine learning, pp. 8748-87 8763. PmLR, 2021. Cited on page 4.
Matthew Roughan, Albert Greenberg, Charles Kalmanek, Michael Rumsewicz, Jennifer Yates, and Yin Zhang. Experience in measuring backbone traffic variability: Models, metrics, measurements and meaning. In Proceedings of the 2nd ACM SIGCOMM Workshop on Internet Measurment, pp.
91-92, 2002. Cited on pages and 24.
Arjun Roy, Hongyi Zeng, Jasmeet Bagga, George Porter, and Alex Snoeren. Inside the social network's 's (datacenter) network. In Proceedings of the 2015 ACM Conference on Special Interest Group on Data Communication, pp. 123-137, 2015. Cited on page 22.
Clayton Sanford, Bahare Fatemi, Ethan Hall, Anton Tsitsulin, Mehran Kazemi, Jonathan Halcrow, Bryan Perozzi, and Vahab Mirrokni. Understanding transformer reasoning capabilities via graph algorithms. Advances in Neural Information Processing Systems, 01778820-788700 2024. Cited on page 21.
15

---

Published as a conference paper at ICLR 2026
Alexander Schrijver. On the history of the transportation and maximum flow problems. Mathematical programming, 91(3):337-44 445, 2002. Cited on pages and 20.
John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov, Proximal policy
optimization algorithms. arXiv preprint arXiv:1707.06344 2017. URL https : ///qittub
com/PredWaad DOTEEE/DDT///tt tree / wwwpaaaaaaseeeeeeeeeeeepeeeeeeae www.aasessess... Cited on page 25.
Lingfeng Shen, Aayush Mishra, and Daniel Khashabi. Position: Do pretrained transformers learn in-context by gradient descent. In Proceedings of the 4 International Conference on Machine Learning, volume 235, pp. 44712 44740, 2024. Cited page 21.
Chengshuai Shi, Kun Yang, Jing Yang, and Cong Shen. Transformers as game players: Provable in- context game- capabilities of pre-trained models. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. Cited on page 21.
Robert H Shumway and David S Stoffer. Arima models. Time series analysis and its applications: with R examples, pp. 75-163. Springer, 2017. Cited on page 44.
Richard S Sutton, David McAllester, Satinder Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems, 12, 1999. Cited on pages 6, 33, and 36.
Jiabin Tang, Yuhao Yang. Wei Wei, Lei Shi, Lixin Su, Suqi Cheng, Dawei Yin, and Chao Huang. Graphgpt: Graph instruction tuning for large language models. In Proceedings of the 47th International ACM SIGIR Conference on Research and Development in Information Retrieval, pp- Cited on page 21.
491-500, 2024.
Gemini Team, Petko Georgiev, Ving Ian Lei, Ryan Burnell, Libin Bai, Anmol Gulati, Garrett Tanzer, Damien Vincent, Zhufeng Pan, Shibo Wang, et al. Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context. arXiv preprint arXiv:2403.055300 2024. Cited on page 20.
Claudia Tebaldi and Mike West. Bayesian inference network traffic using link count data. Journal of the American Statistical Association, 93(442))557-573, 1998. Cited on page 24.
Steve Uhlig, Bruno Quoitin, Jean Lepropre, and Simon Balon. Providing public intradomain traffic matrices to the research community. ACM SIGCOMM Computer Communication Review, 36(1) 83-86, 2006. URL https: //totem./////tt../ 23.
info www. ac..l.c...o. www.aac....aeeeee.........ee..e. dataset .tml.. Cited on page
Asaf Valadarsky, Michael Schapira, Dafna Shahaf, and Aviv Tamar. Learning to route. In Proceedings of the 16th ACM workshop on hot topics in networks, pp. 185-191, 2017. Cited on pages 1, 7, 20, and 25.
Hao Wang. Haiyong Xie, Lili Qiu, Yang Richard Yang. Yin Zhang, and Albert Greenberg. Cope: Traffic engineering in dynamic networks. In Proceedings of the 2006 conference on Applications, technologies, architectures, and protocols for computer communications, pp. 99-110, 2006. Cited on pages 1, 3, and 20.
Heng Wang, Shangbin Feng, Tianxing He, Zhaoxuan Tan, Xiaochuang Han, and Yulia Tsvetkov. Can language models solve graph problems in natural language? Thirty-seventh Conference on Neural Information Processing Systems, 2023. Cited on page 21.
Siwei Wang, Yifei Shen, Shi Feng, Haoran Sun, Shang- Hua Teng, and Wei Chen. Alpine: Unveiling the planning capability of autoregressive learning in language models. In The Thirty-eighth Annual Conference Neural Information Processing Systems, 2024. Cited page 21.
Yufei Wang and Zheng Wang. Explicit routing algorithms for internet traffic engineering. In Proceedings Fight International Conference on Computer Communications and Networks (Cat. No. 99EX370), pp. 582-588, IEEE, 1999. Cited on pages 1 and 20.
Taylor Webb, Keith J Holyoak, and Hongjing Lu. Emergent analogical reasoning in large language models. Nature Human Behaviour, 1526-1111551 15)
2023. Cited on page 21.
16

---

Published as a conference paper at ICLR 2026
Edward Weiner. Urban transportation planning in the United States. Springer, 1987. Cited on page 20.
Wann-Ming Wey and Kuei-Yang Wu. Using anp priorities with goal programming in resource allocation in tramsportation. Mathematical and computer modelling, 46(7-8):985-1000, 2007. Cited on page 20.
Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning, Machine learning, 8(3):299-26,, 1992. Cited on page 41.
Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, et al. Huggingface's transformers: State-of-the-arr natural language processing. arXiv preprint arXiv:1910.03771, 2019. Cited on page 27.
Duo Wu, Xianda Wang, Yaqi Qiao, Zhi Wang, Junchen Jiang, Shuguang Cui, and Fangxin Wang. Netllm: Adapting large language models for networking. In Proceedings of the ACM SIGCOMM 2024 Conference, pp. 661 168. 2024a. Cited on page
Xixi Wu, Yifei Shen, Caihua Shan, Kaitao Song, Siwei Wang, Bohang Zhang, Jiarui Feng, Hong Cheng, Wei Chen, Yun Xiong, and Dongsheng Li. Can graph learning improve planning in L.LM-based agents? In The Thirty-eighth Anmual Conference Neural Information Processing Systems, 2024b. Cited on page 21.
Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, and Philip S Yu. A comprehensive survey on graph neural networks. IEEE transactions on neural networks and learning systems, :2(1))4-22 24, 2020. Cited on page 3.
Yang Xiao, Jun Liu, Jiawei Wu, and Nirwan Ansari. Leveraging deep reinforcement learning for traffic engineering: A survey. IEEE Communications Surveys Tutorials, 23(4):2064 2097, 2021. Cited on page 20.
Zhiying Xu, Francis Y Yan, Rachee Singh, Justin T Chiu, Alexander M Rush, and Minlan Yu. Teal: Learning-accelert accelerated optimization of wan traffic engineering. In Proceedings of the ACM SIGCOMM 2023 Conference, pp. 378-393, 2023. Cited on pages and 32.
Zhiyuan Xu, Jian Tang, Jingsong Meng, Weiyi Zhang, Yanzhi Wang, Chi Harold Liu, and Dejun Yang. Experience-driven networking: deep reinforcement learning based approach. In IEEE INFOCOM 2018-IEEE conference on computer communications, pp. 187 1871-1879, IEEE, 2018. Cited on page 20.
Huimin Yan, Meng Li, and Xi Lin. Time-dependent on-street parking planning in a connected and automated environment. Transportation Research Part C: Emerging Technologies, 142:103745. 2022. Cited on page 20.
Jin Y Yen. Finding the k shortest loopless paths in network. management Science, 17(11):122-716. 1971. Cited on page 27.
Xinyu Yuan, Yan Qiao, Zonghui Wang, Meng Li, and Wenzhi Chen. Lmte: Putting the "reasoning" into traffic engineering with language models. arXiv preprint arXiv:2602.00941, 2026. Cited on pages and 20.
Junjie Zhang, Minghao Ye, Zehua Guo, Chen- Yu Yen, and Jonathan Chao. Cfr-rl: Traffic engineer- ing with reinforcement learning in sdn. IEEE Journal on Selected Areas in Communications, 38 (10):2249-22) 2259, 2020. Cited on page 20.
Renrui Zhang, Dongzhi Jiang, Yichi Zhang, Haokun Lin, Ziyu Guo, Pengshuo Qiu, Aojun Zhou, Pan Lu, Kai-Wei Chang, Yu Qiao, et al. Mathverse: Does your multi-modal Ilm truly see the diagrams in visual math problems? In European Conference on Computer Vision, pp. 169-186. Springer, 2024. Cited page?. 2.
17

---

Published as a conference paper at ICLR 2026
Yin Zhang, Zihui Ge, Albert Greenberg, and Matthew Roughan. Network anomography. In Proceedings of the Sth ACM SIGCOMM conference on Internet Measurement, pp. 30-30, 2005. Cited on
URL https: wwww www.aaaaa........... http:///haaareeerrc//bbb.eeeeeee page 23.
Tian Zhou, Peisong Niu, Liang Sun, Rong Jin, et al. One fits all: Power general time series analysis by pretrained Im. Advances in neural information processing systems, 36:43322 43355, 2023. Cited on page 21.
Zihao Zhou and Rose Yu. Can LLMs understand time series anomalies? In The Thirteenth International Conference on Learning Representations, 2025, Cited on page 21.
18

---

Published as a conference paper at ICLR 2026
Appendix
TABLE OF CONTENTS
A Related Works
20
Detailed Experimental Setup
B.1 MCF Objectives
21
B.2 Experimental Datasets
22
B.3 Baselines
25
B.4 Candidate Path Selection
27
C PRAM Implementation
27
C.1 Prompt Structure
28
C.2 Sub-image Plotting
28
C33 Adaptation Algorithm
29
C,4 Framework Details
31
D Theoretical Results
33
D.I Notations & Definitions
33
D.2 Assumptions & Useful Lemmas
34
D.3 Proof of Theorem
39
D.4 Proof of Lemma
40
D.5 Proof of Theorem 2
41
E Additional Empirical Results
43
E.1 Impact of MLM's Backbone Model
43
E.2 Impact of Number of MLM's Layers
43
E.3
Baseline Results w.r.t. Prediction .e..............................
44
E,4
Results w.r.t. Synthetic Data Distributions
44
E.5 Examples of MLM's Visual Understanding
47
E,6 Results w.r.t. Subgraph Processing Schemes
47
Limitations Future Works
G Usage of LLMs
51
19

---

Published as a conference paper at ICLR 2026
ATOIE WOEES
APOICC COROR
Figure 12: Submarine cables serve as choke points in large-scale topologies for network communica- tion, providing a critical yet expensive infrastructure; the figure is drawn from this URL..
A RELATED WORKS
Solving ColtioCommodity Flow Problems The multi-commodity flow (MCF) is very classic resource allocation problem widely found in various fields such as network (Wang & Wang, 1999; Applegate & Cohen, 2003; Jain al., 2013; Kumar al., 2018; Abuzaid et al., 2021; Krishnaswamy et al., 2022), power (Blaauwbroek et al., 2015; Chalvatzis et al., 2019), transportation (Weiner, 1987; Schrijver, 2002; Wey & Wu, 2007: Noor-A A-Rahim al., 2020). logistics (Mandell, 1991; Balcik et al., 2014; Yan et al., 2022), could services (Chang et al., 2010; Akhter & Othman, 2016) and so on, playing a critical role in service and provider infrastructures. This part will primarily discuss works related to network traffic engineering (TE), which is the most extensively studied MCF problem. For the adaptive TE, LP solvers such as Gurobi are faster in practice, perhaps because they take larger steps towards the optimal allocation, but are still slow for the problem sizes considered here. Unlike adaptive TE, oblivious routing often optimizes worst-case link utilization across all possible demand matrices (DMs) (Racke, 2002; Applegate & Cohen, 2003; Azar al., 2003). To fit current demands and react to failures, SMORE (Kumar et al., 2018) proposed semi-oblivious routing by dynamically adapting sending rates. An improved work is COPE (Wang et al., 2006), which achieves optimization within the space spanned by historical DMs without compromising worst-case performance bounds. However, this combined approach of generalizing oblivious routing and performing optimization incurs substantial computational overhead (especially for large-scale systems shown in Figure 12). Although NCFlow (Abuzaid et al., 2021) and POP (Narayanan et al., 2021) significantly speed up coltii-commoiity flow computations via problem decomposition and parallelization, their methods inherit the limitations of prediction-depender approaches. Nowadays, researchers have also explored ML-based allocation (Xiao et al., 2021). These methods typically use demand-prediction and RL approaches (Valadarsky et al., 2017; Xu et al., 2018; Zhang et al., 2020; Hope & Yoneki, 2021; Bernárdez et al., 2021). For example, CFR-RL RL (Zhang al., 2020) learns a policy to select critical flows for each given DM automatically. To explore the fcasibility of combining GNNs with DRL, GDDR (Hope & Yoneki, 2021) and MARL-GNN (Bernárdez et al., 2021) transformed TE into online optimization problems over graphs to handle topologies of various structures. They represent the state-of-the-art recently, yet still exhibit obvious limitations-it often comes with high computational complexity and sensitivity to parameter settings. More recently, LMTE (Yuan et al., 2026) leverages large language models to solve TE problems, but it was not intentionally designed or optimized for large-scale systems. Compared to the above methods, PRAM is not only easy to train, but also several orders of magnitude faster than them when solving large-scale MCF problems.
Investigating Reasoning Ability of Language Models Recent advancements in large language models have exhibited exceptional proficiency across diverse NLP tasks. Commercial LLMs such ChatGPT (Achiam et al., 2023) and Gemini (Team al., 2024) currently embody the state-
20

---

Published as a conference paper at ICLR 2026
of-the-art in reasoning capabilities. Webb et al. (2023) first claimed that LLMs like GPT-3 have acquired an emergent ability to find zero-shot solutions to a broad range of analogy problems. Gonzälez & Nori (2024) extended it beyond mathematics to vision domains. And Lin et al. (2024) investigated the application of in-context learning (ICL.) in pre-trained models for decision-making tasks, while Shi al. (2024) explored its efficacy in game-playing scenarios. For theoretical analysis of LMs' reasoning capabilities, Wang et al. (2024) shows that Transformers learn planning via nath-finding by encoding adjaceney/reachabbiiy matrices. Subsequently, Sanford et al. (2024) sharply separates Transformers from other architectures and provides width and depth separations via graph algorithms. Moreover, Li et al. (2024) demonstrates that chain-of of-tnooo-thhhhht (CoT) enables bounded-deptl transformers to solve serial computation problems, and Chen & Zou (2024) studied the capabilities of the transformer architecture with varying depth. Huang et al. (2025) proved that a transformer with CoT prompting can learn to perform multi-step gradient descent autoregressively. Several works have also established the existence of deep transformers capable of implementing gradient descent across different domains (Akyürek al., 2023; Ahn et al., 2023; Mahankali et al., 2024; Nichani et al., 2024; Shen et al., 2024). In the theoretical part of our study, we also leverage the MLMs' advanced reasoning power to guide and assist in resource allocation.
Adapting Language Models for TS or Graph Here, we review previous work that employs LLMs for time series (TS) and graph applications. The first category is to boost time series related tasks (Jin et al., 2024b), such as forecasting, imputation, and anomaly detection. To adapt LLMs for time series forecasting, Chang et al. (2023) employed a two-stage approach: fine-tuning GPT-2's transformer module and redesigning positional encoding. Besides, Zhou et al. (2023) propose a parameter- efficient adaptation method for pre-trained language models, named "One Fits All", that preserves their original self-attention and feedforward architectures. Time-LLMM (Jin et al., 2024a) further proposed an embedding reprogramming technique to align the language model's word embedding space with time-series representations. Moreover, Zhou & Yu (2025) conducted a comprehensive investigation into LLMs' understanding of time series data and their anomaly detection capabilities. The second category encompasses graph tasks, which typically include graph learning and graph algorithms. For node classification tasks in text-attributed graphs, GraphGPT (Tang et al., 2024) processed the input graph through a GNN encoder for tokenization before LLM integration. Following this paradigm, MolCA (Liu et al., 2023) applied similar structures when predicting their properties. To enable L.MSs to reason over graph-structur NLGraph (Wang et al., 2023) and Talk like Graph (Fatemi et al., 2024) introduced natural language translation step, followed by few-shot prompting or CoT techniques for effective inference. LLaGA (Chen et al., 2024) reformatted the center node and its neighborhood into a structure-awan textual representation, and Wu et al. (2024b) conducted a pioneering investigation into graph-learning methodologies for task planning in language agents. Unlike existing approaches, this paper advances MCF solutions as a novel framework to bridge the gap between these two research trajectories.
encoders to handle molecular data,
B DETAILED EXPERIMENTAL SETUP
In this section, provide the detailed experimental setup ensure clarity of our paper. We first outline the objectives of the experiments, followed by the datasets used in our study. We then introduce the baseline methods for comparison and, finally, describe the path selection strategy.
B. MCF OBJECTIVES
Using the notations introduced in $ 2, we define three optimization objectives follows.
(1) Maximum Link Utilization (MLU): It refers to the maximum value of all link utilization ratios in the topology, which is classical allocation objective. A lower link utilization suggests greater resilience. Equation (1) summarizes the formulation.
max Le aEE Ce
minimize
subject to fe ΣΣ Da. Tp:
EEE
,VEVV PEP c3p
(1)
Pp 0, Vs,LED E
Σ rp 1.0, Vs
PEP...
21

---

Published as a conference paper at ICLR 2026
(a) Meta DB
(b) Meta WEB
(c) Abilene
(d) CERNET
(c) GÊANT
Figure 13: Five real- real-world network topologies of various systems.
(2) Maximum Total Flow (MTF): The objective computes policy that satisfies the demand and capacity constraints while maximizing permissible flow. We here additionally define pair-wise capacity up that represents the maximum permissible flow between along the tunnel p. up is also an optimizable component of the configuration. Then the optimization problem is formulated as
follows.
ΣΣ fp
maximize
A.EVPEP...
subject to
fp v De,t. Tp: Vs,t V, P.,
fp up: V, Vp P.,
(2)
Pp 2 O, Up 2 0, f0 Vs, Vp P.,
Σ Σ 4p Ce: YeEε
5.EEV PEP c3p
Pp O, Vs vp E Pa.
(3) Maximum Concurrent Flow (MCF): In maximum concurrent-flww problems. the objective is to compute a configuration that maximizes the total network throughput. Specifically, it depends on surrogate bound a, which is an a-fraction of each D.,; is routed concurrently. By defining S(P) as the actual commodity flow the tunnel p between and I, the surrogate problem can be formally expressed as
maximize subject to
a E [0, 1]
Σ
Sal 2 D.,i Vs,teV
PEP
( ddee,, VEEE s.EVV pEP.. cBp
Σ
((p] 2 0, Vs,LEV, Vp E Pa.
f(0) < DT Vs,t E V, Vp E Ps,:
Tp 2.2. O. V, Vp E P...
However, since in many cases the minimum concurrent flow is either very close to zero or exactly zero and thus not representative, in the experiments we instead use the average satisfaction ratio across all flows to capture the intended fairness effect.
B.2 EXPERIMENTAL DATASETS
In this subsection, we provide detailed information on the datasets used in all experiments. Note that all these datasets and topologies are publicly available, and their corresponding access links can be found in the references.
B.2.1 WORLD DATASETS
We first consider two small-scale social network datasets, Meta DB and Meta WEB (Roy et al., 2015). The Meta DB cluster consists of MySQL. servers that store data and process SQL. queries,
22

---

Published as a conference paper at ICLR 2026
Topology
Nodes
Links
# Total DMs
Granularity
Collection Time
Meta DB
4
12
10,000
Meta WEB
8
56
10,000
Abilene CERNET
12
30
48,384
minutes
March ~ September 2004
14
32
10,000
5 minutes
February ~ March 2013
GÉANT
23
74
10,733
15 minutes
January ~ April 2004
Table 1: Real-world network topology statistics
GtsCe (149, 386)
(b) Colt (153, 354)
(c) UsCarrier (158, 378)
(d) Cogentco (197, 486)
(e) (754, 1790)
Figure 14: Five large-scale topologies used in evaluation.
while the Meta WEB cluster handles web traffic. Both clusters operate at the Point of Delivery (PoD) level, which is fully connected. We also use three real-world WAN datasets: US Internet2 Network (Abilene, Zhang al. (2005)), Pan-European Research Network (GÉANTI Uhlig al. (2006)) and China Education and Research Network (CERNET, Li et al. (2011)), each with public DMs captured different snapshots. Table presents key statistics (i.e.. number of nodes, links, and samples) for these datasets. Besides, we visualize their topologies in Figure 13. All DMs are normalized by ten times the maximum link capacity of their topologies for DNN training and testing (hereinafter the same).
B.2.2 SYNTHETIC DATASETS
To evaluate PRAM's ability to scale to larger systems, we also extracted several topologies from the Internet Topology Zoo (Knight et al., 201 They are GtsCe with 149 nodes and 386 edges, Colt with 153 nodes and 354 edges, UsCarrier with 158 nodes and 378 edges, Cogentco with 197 nodes and 486 edges, and Kdl with 754 nodes and 1790 edges. Figure 14 shows visualizations for the five used topologies; note that we set 1,000 for all link capacities. Also note that we only take the largest strongly connected component in these topologies for evaluations.
23

---

Published as a conference paper at ICLR 2026
We synthesize DMs for the above large topologies based on three typical types of distribution or method' All datasets except for Kdl, the number of generated samples 3,000 while that of Kdl is 500 due to its super-high dimension. We present their details as follows:
Gravity (Roughan al., 2002): The distribution has been an empirical success in that it accurately predicts trade flows between countries for many goods and services. Its key idea is that the total demand leaving a node proportional to the total capacity on the node's outgoing links; this demand is divided among other nodes proportional to the total capacity on their incoming links. We present the Python pseudocode for the corresponding synthetic data in Algorithm Specifically, the algorithm first computes the total inbound and outbound capacities for all nodes. It then derives the DMs by sampling from a Gaussian distribution centered the expected fraction, with variance proportional to the same value. Finally, the resulting demand matrix is rescaled by factor to match the desired total resource level.
Algorithm 1 Gravity-based Demands Generation in a Python style
G: graph or topology scaling factor
num_nodes -
len (G. nodes) f number of nodes in_cap, out_cap = (), {)
A = zeros nun_nodes, num nud_nodes) # initialize DMs
for u in G.nodes:
in_cap [u] = sum (v, u) for v in predecessors (u)
out_cap[ul [u] sum (cap (u, v) for v in successors (u))
for in n.dde
norm_u out_cap [u] / sum (cut_cap) for v in n.deds::
frac = norm_u in_caplvl (sum(in_ (in_cap)
$ allocate resource with randemness A[u, v] = max (Gaussian (frac, frac/4), 0)
r scale by constant facter
Poisson (Tebaldi & West, 1998): This model can be formally expressed as P(A, 8), where the demand between nodes and follows Poisson random variable with mean A8da... Here, da,t denotes the hop distance of the shortest path between s and and E [0, is a decay factor controlling demand concentration. Choosing close to 0 yields highly concentrated demands (favoring nearby nodes), while values close generate more uniform demands across the topology. The corresponding Python-ste wython-style pseudocode is given in Algorithm 2. The algorithm first computes the all- all-pairs shortest path distances. For each source-destination pair, the expected demand set to Add..., which is then sampled from Poisson distribution to incorporate randomness. The diagonal entries are zeroed to exclude self-demands. Finally, the entire demand matrix scaled by a factor r to adjust the total resource level.
Bimodal (Applegate & Cohen, 2003): In the bimodal model, fraction p of source destinatio pairs (selected uniformly at random) are assigned demands drawn from Uniform(b, c). while the remaining pairs receive demands from Uniform(O, a). This construction creates heterogeneous demand matrix where some pairs experience significantly higher traffic than others. Algorithm provides Python-style pseudocode for generating synthetic data under this model. The algorithm first initializes an empty demand matrix. It then uses random assignment to decide the group for each entry. Depending on the group, entries are sampled from the corresponding uniform distribution. Finally, the diagonal entries are set to zero. This yields a demand matrix with a bimodal distribution of traffic intensities.
'The code mainly based this Github repository: http:: hwt./net.contrratt/eeeee.
24

---

Published as a conference paper at ICLR 2026
Algorithm 2 Poisson-basee Demands Generation in a Python style
graph or topology
lam: base Poisson parameter decay factor (<-1)
decay:
scaling factor
num nodes
Len (G.ndees)
number of nodes
compute all-pairs shortest distances distances zeros (num_nodes, num_nedes) for src, dist_dict shortest_pach_legghhii for target, dist in cistances(sre, target] dist
:):
generate Poisson-based demand matrix array (poisson (lam * (decay dist)) for dist row] for row in distances), dtype=float) fill_llaa ciagonal (A, 0.0) # no demand for self
scale by constant facter
Algorithm 3 Bimodal- based Demands Generation a Python style
G: graph tepology fraction; fraction of low-range allocations low_range: [min, max] for low allocations range: [min, max] for high allocations
(G.nnee) & number of nodes
initialize demand matrix
A zeros (num_nodes, num_nodes)
p= [fraction, 1 fraction]
randomly choose low/high demand for each entry
inds = random_choice___11 (num_nodos, num_nodes) with probablity of
low allccations
[inds]] uni form (low_range lew_range[1] [1], sum (inds)
high allocations
Al-inas| uniform range (0), range [1], sum (-inds))
fill_diagonal (A, 0.0) no demand for self
B.3 BASELINES
In this paper, we compare PRAM against the following state-of-the-art MCF solutions
DRL (Valadarsky et al., 2017): leverages deep reinforcement learning to replace explicit demand prediction with end-to-end optimization, mapping recent DMs to MCF configurations. Specifically, we train policy neural network of the same size as PRAM'S trainable parameters, using OpenAl's proximal policy optimization (PPO) algorithms (Schulman et al., 2017).
HARP (AlQiam et al., 2024): It GNN-based scheme which ensures invariances to natural input transformations (e.g., permutations of node ids, tunnel reordering), and has a neural architecture
p "Implementations of all non-open-source baseline methods can be found in our codebase. For HARP and Acther, follow the settings in their papers and experiments.
25

---

Published as a conference paper at ICLR 2026
Approach
Abilene
GÉANT
UsCarrier
Cogentco
LP-w
19.67 8.26
1.60 0.54
67.88 7.25
73.39 2.87
LP-f
22.36 6.47
2.87 = 1.09
10.08
0.03
17.08 0.06
Table 2: Comparing different optimization scheme for minimizing MLU objective.
aligned to optimization models. HARP has focused on the MLU objective, so we extended the other two objective functions according our framework. Due to its end-to-end nature, this extension is totally fine. Also note that different from HRAP's original implementation which uses exact demands as input of models, we here modified it with historical information.
Aether (Fan et al., 2025): It is most recently proposed algorithm, which excels in generalizing across different networks and different amounts of demands. addition to graph Transformer, Aether also employs a differentiated traffic strategy based on multi-agent RL to focus on larger flows for better learning while handling small flows with rules.
LP (Gurobi Optimization, LLC, 2024): It solves the MCF optimization problem for all demands
using linear programming (LP), which Gurobi (version 1 11.0.3) optimization solver is employed. The code for the different MCF objectives is listed in Algorithm 4 to Algorithm 6. We include two variants for minimizing MLU: one that treats path weights as the decision variables (LP-w) and another that optimizes over path flows (LP-f). As shown in Table the formulations yield solutions of noticeably different quality. LP-w performs reasonably well on small topologies, but its effectiveness drops sharply at larger scales. Given the stability and consistency of the results, adopt LP-f as the default baseline throughout this paper.
POP (Narayanan et al., 2021): It also decomposes large-scale granular flow problems into subprob- lems that are solved in parallel using an LP solver. Specifically, the entire topology is replicated k times, with each replica assigned & of the original link capacities. Commodity demands are then randomly distributed across these replicas, and each subproblem is solved independently in parallel. 2 for the five real-world datasets, & for four
set i according to the topology size: large-scale topologies other than KDL, and k 128 for KDL.
LP-top (Namyar et al., 2022): It implements a simple yet effective heuristic algorithm that is recently revealed "demand pinning". It allocates the top 10% of demands using an LP solver and assigns the remaining demands to the shortest paths.
(pred): Since most of the non-learning method (i.e., LP, POP, and LP-top) optimizes the solution with perfect knowledge of future demands, we consider their practical implementation based on predicted future demands. For simplicity and efficiency, primarily use non-parametric weighted moving average for prediction. Experiments on other prediction schemes are presented in Appendix E.3.
Algorithm 4 Optimize weights for MI.U in Gurobi style
E: set of edges, each with capacity c[e] PathToBdge [p, e] 1 if path uses edge e
create optimization model Model (); mlu m. addVar
for each pair and path k in (s,d) add variable w[a,d,k] in [0,1]
add constraints
for each (s, d): sum_k w[s,d,k] for each demand D and each edge e in E: load = sum over ks) d[s,d,k] D[s,d] v PathToRdge I (s, d,),, el load <= mlu + Clel
minimize maximum 1ink utilization m. setobjectiiv (mlu); m.optimize()
26

---

Published as a conference paper at ICLR 2026
Algorithm 5 Optimize flows for MLU in Gurobi style
set of edges, each with capacity Cle]
PathToEdge [p, el 1 if path p uses edge e
create optimization model Model (); mlu m.addVar (1b=0)
(p: m.addaar (1b=0) for p al1 paths)
add constraints for each edge
load sum over p: f(p] PathToEdge [p, e]
load <= mlu A c[e]
for each commodity k: sum over in P[k]: : [[p] == D[k]
minimize maximum link utilization
m, setobjective (mlu); m.optimize()
Algorithm Optimize flows for MTF MCF a Gurobi style
E: set of edges, each with capacity C[e] PathToEdge [p, e] 1 if path p uses edge
create optimizaiion model
m Model (); f (p: a.addVar (1b=0) for in all_paths)
if objective "MIF"; maximize total delivered flow obj sum over all p: f[p]
else:
maximize concurrent flow facter alpha
alpha m.addVar (1b=0, ub=1)
for each commodity k: sp in P[k] [[p] >= alpha D[k]
obj = alpha
edge capacity constraints for each cdge e: load sum over p: f[p] + PathTcRdge [p, e]
load <= C[e]
per-commodity flow cannot exceed demand for each commodity k: sp PIklI flp] dik]
opt aptimize
je (obj, maximize-True), opoptmmize
B.4 CANDIDATE PATH SELECTION
For each topology, unless otherwise specified, use Yen's algorithm (Yen, 1971) precompute up to shortest paths between every pair of nodes, which serve as candidate paths for multi-commodity flow allocation. fewer than paths exist, the last available path is repeated to ensure a total of
C PRAM IMPLEMENTATION
We implement PRAM using PyTorch (Paszke et al., 2019) and HuggingFace Transformers (Wolf et al., 2019). Most experiments are conducted on a Linux server equipped with two NVIDIA A100 GPUs (80GB memory each). And adopt 7:1:2 ratio for training, validation. and testing across all datasets. For model initialization, we employ Kaiming uniform initialization for the low-rank matrices. Additional implementation details are provided below.
27

---

Published as a conference paper at ICLR 2026
Subtask Prompt Structure
## Role You are an expert in solving multi-commodity flow problems.
## Task : Based on [lviiioo__stere |> iimage_ __dd>p//diiiooo Lirion__dddl>_ the all traversed links from source node (marked in red) to others the image and other information, determine candidate path weights for each dourie-destination pair in the near future. Current Source node need to optimize <i>
## Goal The objective is to minimize <maximum link utilizaatin>> Fiidd set of candidate pa- th weights (split ratios) that minimizes the highest utilization across all links the network.
## Description: There are <num node_? nodes in the topology and <num_paths> shortest paths betw- een each node pair are used as condidate paths. Note that the sum the candidate path weights should equal 1, All commodity demands mu-st be routed (conservation of flow), while weights must be positive floats. ...... (other descriptions)
The most recent demands from (<i>, 0) to (si>, rmum_nodees 1) are as follows. An <XL, x_2, x, and history mean demands from (<i>, 0) to (4i>, <num_nodes> 1) are as follows. \n <mL,,
## Demands:
m_2, "., m (other statistics)
follow the answer template. [source 0 --pttht-weight, source-O Source-O-pathx- weight], [source-1-pathie 1-tthh--weighhht, ...source---pppe 1-pathx-weighh),,.. And no allocation is needed for the sour- ce node to itself.
## Format Make
Figure 15: Example subtask prompt structure for maximum link utilization optimization.
C.1 PROMPT STRUCTURE
We show the overall prompt structure used by PRAM for minimizing maximum link utilization in Figure 15. Our design can be roughly divided into three parts: task instructions, domain knowledge, and commodity information, the MLM as an expert in multi-commodity flow problems, ensuring that subsequent reasoning is guided by subtask-oriented perspective. The task and goal sections clearly specify the optimization target-mininizing maximum link willization-wiile constraining the solution to path weights that obey feasibility requirements (e.g., negaatiity and normalization); (ii) Domain knowledge: The description encodes domain knowledge, such as the number of nodes, candidate paths, and flow conservation principles, thereby narrowing the search space toward admissible allocations; (iii) Commodity information: The demands section conveys the actual commodity information, including most recent traffic volumes and their historical statistics, enabling the model to reason about both instantancous and smoothed demand dynamics. Finally, the format clause enforces deterministic answer template, which facilitates automatic parsing and validation of the policy head network. The template for the maximum flow and concurrent flow is essentially the same the outlined above.
illustrated below. (i) Task instructions: The role section positions
Algorithm 7 Draw sub-images per source in a Python style
network graph
group by source grouped_paths (s: all paths starting from s}
for
8, paths in subG = combine all edges in paths layout (subG)
# compute laycut positions and scales draw all nodes and highlight scurce node
draw all edges, add arrows if is directed for edge in subG.edges: draw edge label G[edge] [' ['capacity")
C.2 SUB-IMAGE PLOTTING
We utilize Matplotlib (Hunter, 2007) and NetworkX (Hagberg et al., 2008) to generate visual representations of the sub-graph data. We draw all candidate paths grouped by source node. Each
28

---

Published as a conference paper at ICLR 2026
Figure 16: Examples of input sub-images on GÉANT topology. In each image, the starting point of the commodity flow is marked in red, only the links that exist in the candidate paths are drawn, and the link capacities are directly indicated on the links.
source gets figure containing all its candidate paths (to all destinations). The pseudo-code is presented in Algorithm 7. These visualizations are then provided to multimodal LLMs capable of processing image inputs (.png file with 240 dpi). Figure 16 shows all sub-image examples from the GÉANT topology, where its source node is marked in red and the link capacities explicitly indicated in the figure.
C.3 ADAPTATION ALGORITHM
We next provide implementation details that complement the main text. We first outline how objectives (i.c.. MLU, MTF and MCF) are computed efficiently as rewards using parallel matrix operations, followed by the multi-agent RL procedure adopted to fine-tune PRAM.
C.3.1 OBJECTIVE (REWARd) COMPUTATION
For MLU, we train PRAM to output all path weights (with Sigmoid non-linear as its final layer). The mapping between these weights and the resulting MI.U can be represented using Algorithm 8. It is
29

---

Published as a conference paper at ICLR 2026
noteworthy that our reward computation relies real samples (demand future time steps), whereas the model only has access to historical information. No need for any loop operations, this mapping can be established through simple matrix operations in parallel. Regarding MTF and MCF, force PRAM to directly predict allocated demand Wp for each candidate path (with ReLU non-linear as its final layer). As seen in Algorithm 9, we then scale w/p ? where y = max mmxxae Eprepp ate) 1) In this way, PRAM guarantees that link capacity can be exceeded, and the computation is also fast.
Algorithm 8 Compute MLU in PyTorch style
path weights (split ratios)
d: commodi ty demand vector
# C: 1ink capacities
PathtoEdge: path-to-edge incidence (1 if edge j in path i else 0)
SDtoPath: SD-to-path incidence (1 if path serves conmodity i else
normalize path weights total_w = SDtoPath 8 w = w (SDtoPath.T @ (1.0 / total_w))
demand en paths and flow on edges demand_on__athin = (SDtoPath.T d.I) w Elow_on_edges = P @ demand paths
maximum 1ink utilization congest on flav_on_edeee C MLU max (congestion)
Algorithm 9 Compute MTF & MCF in a PyTorch style
w_p: predi cted demand vector
d: commodi ty demand vector
C: 1ink capacities
PathtoEdge: path-to-edge incidence (1 if edge j in path i else 0)
SDtoPath: SD-to-path incidence (1 if path serves SD else 0)
scale planned demands on each path w_p_scaled w_p / max ( ((PathtoEdge.1 T @ w_p) / C)
compute flow per SD pair, limited by demand flow min ( (SDtoPath w_p_scaled) T, d)
total flow MTF sum (flow)
maximum concurrent ratio MCF min (flool[ !- 0] / did !- 01)
C.3.2 FINE-TUNING DETAILS
As detailed the main text (S 3.3), PRAM opts for employing the adaptation framework of multi-agent RL. Since MCF allocations are deterministic, modeling To (a, as a Gaussian distribution: during training, actions are sampled from this distribution, while at deployment the mean is used as the deterministic allocation. To be more specific, we employ a separate one-layer MLP to predict the mean, while keeping the standard deviation fixed". After sampling from the corresponding Gaussian distribution, the log proba-
adopt the standard approach of
bility is calculated as: log mo(ag)s;) - [log(2mo7) (m=="))) which is implemented by
We also experimented with learnalle standard deviations in our early trials, but the results suhoptimal.
30

---

Published as a conference paper at ICLR 2026
cicchhdistibbtiins..n normaal.o......o.g..o.o.oooo.... hatpaaa.oooo.oobb......... Algorithm 10 presents the pseudo-code for fine-tuning PRAM. leverages the observation that allocations in MCF problems computed for a single time interval do not influence subsequent intervals (e.g.. the demand matrices). This domain-specific insight enables us to simplify training by reducing the long-term return to a one-step return. Furthermore, note that the subproblem decomposition and the -uug image transformation are performed only once the beginning of the task, ensuring that they do not compromise the efficiency of model fine-tuning.
Algorithm 10 Fine-tuning PRAM in PyTorch style
Model: policy network (pased on MIM) Optimizer: optimizer for Model
episodes: sequence of training episodes
for observation in loader:
observation: pre-processed sub-images and text prompts loss =
for step in episode:
action, log_probability = Medel (observation) select action
advantage compute_object (actiionn
(action) get multi-agent reward
# accumulate policy gradient
loss +- (- 1og__rrbabbliity advantage) mean
Opt
loss. boss.backward
oetimizer.stepl
if EarlyStop and converged (): break
C.4 FRAMEWORK DETAILS
In this subsection, we first outline the architectural information of PRAM and provide an overview of our hyperparameter choices.
C.4.1 ARCHITECTURE
As shown in Figure 17, given Owen2.5-VL-style multimodal language model, images inserted at the position marked by "< |vision vision_start] >< |image_padd >< |vision_enda >" (see Figure 15). Text and images arranged in the order they appear in the original prompt. The context prepended to the input as prefix. We apply LoRA fine-tuning exclusively to the attention blocks of the language model, specifically restricted the QV projections. As described in the preceding subsection, the final stage of PRAM'S 's architecture samples actions (i.e., path weights) from Gaussian distribution parameterized by the predicted mean and a fixed standard deviation.
C.4.2 HYPERPARAMETER
Since PRAM primarily relies on the multimodal interface of the pre-trained model, most components (including token embedding dimensions) are fixed, leaving only limited set of parameters tune, as detailed below. First, we set the length of the learnable context proportional to the number of nodes the topology. with minimum of 48 tokens. Second, in the cross-attens noss-attention module, the embedding dimensions of the Q. K. and V matrices are chosen from [64, 128, 256, 512]. Third, we take the final hidden state of the language model as input to the policy head, which consists of single fully connected input layer, followed by two output heads (for mean and standard deviation), each implemented as single fully connected layer (with output size 2 number paths X number of nodes). To speed up processing, For the sampling batch size, we select from [8, 16, 32, 64]. The sampling granularity defined at the cource-node level, meaning that allocations for larger topologies require multiple batches. For example, allocating resources for 150-node topology requires at least three batches. However, also increase the partition granularity by aggregating multiple nodes into larger groups,
crop pre-saved sub-images to 128 128 512 x 512 pixels.
31

---

Published as a conference paper at ICLR 2026
Action (Weight) Distribution
+
Mean
Std
Policy Head
Add & LLayer Norm
Feed Forward
Casual Multimodal LM Backbone
Add & Layer Norm
Context Tokens
Text Tokens
Image Takens
Text Tokens
Multi-Head Attention
N
Cross Attention
Vision Encoder
Learned Context
Law-conk Matcices
Freezing
Trainable
Subgroph os Imagg
Figure 17:
Model architecture of PRAM. In the figure, we use Qwen2.5-VL-style MLM, which integrates of a vision encoder and a language model decoder to process multimodal inputs.
thereby reducing the number subgraphs and the overall computational and memory footprint (to enable non-cross-batch operations). Finally, the learning rate for fine-tuning is selected from [10-3,5 10-4 10-A1, and the adaptation of PRAM is limited to maximum of 10 epochs with early stopping. Moreover, by default, the Adam optimizer (Kingma, 2014) is employed throughout all experiments.
C.4.3 POST-SOLUTION REFINEMENT
Although our MLM- -eo---d method already produces high-quality allocation results, there still room for further improvement under certain test conditions. We refer to this fine mee-grained adjustment as post-solution refinement. When partial demand information or other known conditions are available, post-solution refinement can be carried out in roughly three ways: (i) running gradient descent with ADMM (Xu et al., 2023), (ii) recurrent adjustment with RNNs (AlQiam al., 2024), and (iii) LP. based partial solution refinement (Liu et al., 2025). Herein use the idea of (ii) for our refinement. Specifically, we can select O ((V)) flows with the highest (Gaussian) variation by the policy head as the manageable demands for solution refinement. This approach still reduces the complexity and scale of the MCF problem, making tractable for large-scale scenarios while maintaining optimal or near-optimal performance.
32

---

Published as a conference paper at ICLR 2026
D THEORETICAL RESULTS
In this section, we provide the detailed theoretical proof' that underpin the main results presented in the paper. Our goal is establish rigorous guarantees for the proposed method, including its correctness, convergence, and properties. To this end, we first formalize the necessary assumptions and notations, and then proceed derive the main lemmas and theorems step by step. particular, our arguments borrow insights from classical results on convex optimization and more recent analyses of Lss. Some key inspirations were excerpted from prior works of Bertsekas & Tsitsiklis (1996); Sutton et al. (1999); Foerster al. (2018); Perry al. (2023); Ahn et al. (2023).
D.1 NOTATIONS & DEFINITIONS
Notations Below, we list our notational conventions used in this section.
||||| is the Euclidean norm.
n denotes the total number of nodes, i.e., [V].
Cmin denotes the minimum link capacity.
Cmaz denotes the maximum link capacity.
D Dmax an upper bound on the maximum demand between source-destination pair.
D Dmn lower bound on the maximum demand between source-destination pair. Al- though it can be 0, to ensure the validity of the results, we replace O with value arbitrarily close 0.
p denotes the maximum number of candidate paths interconnecting source-destination pair.
a denotes an allocation configuration, which can be represented as a vector with n² x components. It is noteworthy that each source-destination element in is itself vector whose components sum to < 1, specifying its weights T(-) across at most candidate paths.
Ci denotes the maximum link utilization objective function, which, in its simplest form, is (T) max.gg fe(m)
C, denotes the maximum total flow objective function. For convenience, we reformulate it as follows: Let the allocated flow on each link (T) = Eeep (T((T) where (T)()) is the allocated flow on path p. Then the actual flow on p fT( = (T( min min, -.(T): 1)1 and the actual flow successfully transmitted between and f. is fs. Epepe, Sp((T).
Now, we can define (T( = Σ., [T,((T) or Ex,t min (s,,s
Le denotes the maximum concurrent flow objective function. Similar to Lz, we can also reformulate as follows: fe(T) = min ({ (Lee(T) min ({ [n((*) Dua | > 0) (1)).
D,, 1,,E,,,,.r > 0}) or
R(-) denotes the long-term expected reward per step. Given policy it is defined as so, m]. where 7, 80 and denote discount rate, start state, and reward at time / respectively.
R(()) = E o
-f-1r;
(-)..) is an action-value function. Given a policy n, is defined as Q" = Titk s; = s, 4 = a, T]. where aj is the action at time
E |Etn -
d (.) is the stationary distribution of states. Given a policy n, it is defined as d= E EE
Pr (s = so. x]}.
Definition 1. (Convextty) A set C' in vector space is convex for any vectors u, C and a [0, 1), we have that au + (1 a) v e C. A function f + convex if (au + (1 a) S
( (1 a)) f(v).
Definition 2. (Lipschitzness) Let C' C Rdd, A function f Rd + RR is p-Lipschitz over ifr every
W1, Wa C we have that (w]) f(wa) || pl[wn wzll-
33

---

Published as a conference paper at ICLR 2026
Definition 3. (Subgradients) Let S be open convex set, and f R be convex function. A vector that satisfies f f()) + (u w, u) I f(u) for any u E S, is called a subgradient of f at w. The set of subgradients of f at is called the differential set and denoted (
Definition 4. (Gradient Descent, GD) Gradient descent an iterative optimization method that, at each step, updates the solution by moving in the direction of the negative gradient of the objective function evaluated at the current point, i.e. ((0 = ((t-1) - Vol.
Definition 5. (Markov Decision Process, MDP) MDP is defined by the tuple M AS, The state, action, and reward at each time are denoted 8, E S, at E A and T2 E IK respectively. P(s', S, a) Pr [ rewards E [rt+1 9t S, aq = a). and 7 E [0, 1) the discount factor. The agent's 's decision-making procedure each time is characterized by policy, xo(a (8) = Pr {a+ = a 8t 8, 0), where parameter vector. We also write just π (ss a.) for To(a 8).
= S. = a} denotes the transition kernel, R(s, a) the expected
Hazan el al. (2015)) function f : Rdd + is quasi-convex if. for all y IR with f(y) S f(x), (VI(x).: - x) S 0; or it satisfies that its level sets, (f) = x f(x) S a) are convex sets for all a E R.
Definition
D.2 ASSUMPTIONS & USEFUL LEMMAS
Assumption 1. (Convex Concave Objective) All three objective functions, i.e., maximum link utilization, maximum total flow, and maximun concurrent flow, are convex/concave with respect to the input candidate path weights.
Assumption 2. In each attention block, there exist token embeddings of (z.,, y4) from the training set that are aligned with the embeddings of the learned context.
pairs
Assumption 3. In each attention block, the token embeddings of the observation-deccioo pairs (304, ye) approximately satisfy a linear relation" ys W where W RNNxNN is weight matrix.
Note that the aforementioned Assumption can actually be relaxed slightly; long mon-s quasi-convexity (see Figure 18 on the right) is satisfied, it does not affect the related GD-convergencee results in this paper, by using the study of Konnov (2003) and Hazan et al. (2015) (Theorem 5.1). Next, we explicitly prove the core property for all three MCF objective functions Proposition support our insight.
[19661 M
Lemma 2. a1-(1- bz+(1-z)d 2 min (g: E [0, holds for any
Figure 18: quasi-convex Locally-Lipschiz function.
a.b, d 0.
Proof. Let f(x) aI+(1-r)e then we have its derivative: be+(1=2)d'
f((x) (a c) (bxr (1 r)d) (b d) (ar + (1-x)c)
ad - bc
(bx + (1 x:)d)²
(br + (1 x))d)
Now, since f(0) á. f(1) and f'(x) is monotonic on the interval f). min (g. g).
az -r
bz+ (1-x)d
Proposition 1. The objective function L, is convex, while La and Le are quasi-concave, in the flow allocation configuration.
Proof. We prove the corresponding properties of the three functions sequentially. First of all, according to Defination 6, we shall prove a function f is quasi-concave if ff (Ax + (1 (1- x) 2 min (f (x), ff (z')) for any X E [0, 1].
L, convex. The function (T) is, by definition, linear in the splitting ratios म. Since the & is expressed the maximum of functions that are linear in n,, and the maximum of linear (or convex)
functions is convex, it follows that (T)) max.es f=(x) convex with respect to π. 6a
'Note that we only assume the attention inputs after token embedding are approximately linear, not that the original problem is linear. Indeed, linear representation model (e.g., followed by an ML.) can express any nonlinear function (Ahn al., 2023).
34

---

Published as a conference paper at ICLR 2026
L. is quasi-coneave. We denote by ó the normalization constant 1/ min mine 2.(=).1) 1) max maxe =x(=), ,e(m),1) .1). then L. = E.,t Epep,, *p(m) We also denote zp pp(")) = Axp(T) co
(1- X) x)((')), and let o" denote its corresponding normalization constant, respectively. There- fore, by using Lemma 2,
min (C(()),(((T) = min
ΣΣ 3, PEPs,t
Ip(a))
Ip =p(or) &
93 pEPstt
Ex,t Epep,
Tpp Tp( + (1 A) Ex,t Epe P.,. Fp(T))
AΦ (1 X) Ø
EpePi (Ax (T) + (1 2x(('))))
AΦ + (1 (1- L((TT),
3,
Tx(T")
8"
p.P.
where the second inequality is because
Ie (T(T")
(xe(TT + +(1 1)z 2)
6" =
max
max
= max
max
Ce
Co
(xe((T)
(1 - 2) . .1-2)
< max
max
.
max
max
(4)
Ce
Te (e(
max Fe( Ao + (1 (1-1)0'
= max
max
1
- (1 A) max
Ce
Ce
L. is quasi-concave. To analyze Lo, in the following proof, we reuse some definitions from the proof of C, Similarly, we have
Eperar En(x)
Eperar (x(x')
min (C.( = min
min
min
Da,
Do,
Eper, Zpim)
Eperat
ea(a 6¹
X min
+ (1 A) min
De.t
D..
dØ (1 2)
(1)(+)(1-A)))
ΣpePar A0+
=min
De.t
Epepar =p(x")
S min
LoTT),
Dett
where the first inequality is by Lemma 2, and the second inequality is by Ineqn (4).
Lemma 3. Let A be nonempty open convex set, and let f A - R be convex function. Let 0. Then the following are equivalent: ① f is p-Lipschitz continuous on A. ② At every point w E A, every sub subgradient u E Of(w) satisfies ||a||| p.
Proof. First, we assume that f is p-Lipschitz. Now choose w E A and E af(w). Since A is open, have u
T E A by choosing appropriate 0. We write (u w. v) <||u|| and ||u - ull = c. Using Definition 3, i.e., f(u) 2 (w) + (u w, v), we obtain f(u) - f(w) 2 <||=i|- Recall that fu) fw) S pl|u w|l pc, which means |||||| S p.
Second, we assume that || S p. From the definition the subgradient, f(w) - f(u) S (c, w - u).
Now using Cauchy-Schwand inequality we have (ww) f(u) ||a||| ull S p// u/I. By
35

---

Published as a conference paper at ICLR 2026
repeating similar steps, also we have f(w) S pllw- Therefore, [lf(u)-((w)) - pl w- f is p-Lipschitz.
Lemma 4. Consider a convex, p-Lipschitz
f. Let wt be a vector minimizing f(w), in which Iwlll is bounded by B, and GD algorithm with an update rule of the form w(t+1) wl() nil where
V1 E =.f(w), w(1) = O and ū- = Ei= w()) If we run the algorithm on f for T 2 B² p
2 steps with
B2 7 PT
then the output vector w satisfies f(w)- f((w) S c, <,vc> 0.
Proof. We start by Jensen's 's inequality,
T
f(m) - f(w"(()))) f f ) -((w)) S 1 Σ f (wo1)) - f(ww). T Σ m1())
Thus we have f(w) - ) S Ei= (s (1o(1) - -s(w))) using Definition 3, we can write
f()))(ww") + Σ (f(w0))-((w*))
- -s(w*)
(wi)) w,, vu)
To bound the right-hands side, we focus on (wo(()
- v1). Given the positive strength 17.
( (wll)) - w,, vx) ,,,,), = { 10000 - w,, mmr)))
21 (1 [[u(() ""
|| 1o(0) - w - mmall 1|] + n² |1
-nye
(t) - w" || | - |oo(c+11) w -12 + n² llull))
2n
Then we have that
T
T
(1) - wn, .00)
Σ
(1) - w0
10 (e+1) w0 10=1 + 17² |luel
=
21 m=1
17 T w
1
|
11 (1)
w²
w
(t-1)
1
11
21)
2n
2
[=
=0
20
1
||
112
2n
Now according to Lemma 3, |lvel| S p, and ||00|| S B. Plugging them along with 1) v
B
in the above inequality, we have 1 Σ ( (w - w,, mx) < V Therefore, for all c > 0, after
T > B²p² 2f iterations of gradient descent, the objective function converges, i.e., f(w) f (w") Sc which concludes the proof.
Lemma 5. (Policy Gradient Theorem, Sutton et al. (1999)) For any MDP M AS,A, R,7),
ØR
tt(,, a) Q=
Σ (s)
00
00
Proof. We begin with the following observation:
(=(s,a)= = E
r11 + y
2
retk
= =, = a,
2=
= =E(re+1 St = =, a) + Σ PPr PP = s,
Σ
k 2 reth
8141 = =
=R(s,a) + EETP(s
yP(s' 8, a) Σ T(S a a0 (s)
36

---

Published as a conference paper at ICLR 2026
Defining the value function as (*(s) E., ((..)) S, we have that
OV
(ss,a)
00
00
( a)
Q (s. x
0
Q* (s.a)
00
00
am(s, a)
0
[(s,a) + E)P(6,s,O))T())
Q" (3, (s,
00
00
1m((s ("((sa) + (ss Σ
(v*(3')
Σ
PP(s' 8,0)
00
00
By expanding the recursion iteratively, i.e.,
(v)
(V)
(s")
we obtain
00
00
00
(v*(s)
Ôn (a, a)
ΣΣ
* Pr(s + x.
(z)x)))
00
00
k=0
where Pr(s - x, π) denotes the probability of reaching state from state in exactly & steps under policy иT. Also we have
(*(() = Σ a
Totk 8t = s.a, at =
E
reth 1t =
Lk=1
Finally, the gradient the expected discounted return R follows directly:
OR
8
00
=
00
E
rı
so, π = v (su)
00
Um a)
Σ
* Pr(so
+ s, k.
Q" (s.
06
@T(s)a) a)
Σ
d (s)
Q² (s)a).
00
Hence, the proof is complete.
Lemma 6. Let fw S A + IR be parametric approximation Q (s, a) with parameter If fw aR Σ. d ( Σ.
locally optimal and satisfies @f-(s.0)
@mis.a) then
@n(a.a)
fu(ss a).
86
89
30
Proof. We consider learning for by sampling trajectories under policy ा and updating w via a squared-error loss: Aw, du ((((((ss a.) fa (st, a,)l [ a.) - as)) (f the where Q" (s,, a.) denotes an unbiased estimator of Q (8t, ax). At convergence to local optimum, the update direction vanishes in expectation, yielding
á
(s, 0.
Ed ETss, (s, a) [s)
Ow
Substituting the assumption (5.(s.))
Um(s,a) obtain Σ,d d"h (s) E. (=(s.0) (Q*
Oue
H(s.a
(8. a)] 0. This shows that the approximation error of fu is orthogonal to the gradient of the policy parameterization. Now recall the policy gradient theorem (Lemma 5) and subtract the orthogonality condition above, obtain
aR 00
(s, Q (s,
Zm (s,
Σ
-s
ds)
Σ
(Q" (s)
00
00
aT( (s)
Σ d Σ
09
This establishes the claim.
37

---

Published as a conference paper at ICLR 2026
Lemma 7. Let f be any differentiable function, and wi be sequence generated by GD-style update
rule Wx+1 W, + T where for all t, Vt satisfies ① [ (wi)ll < - (w²) U2i || I
22 VVV (w+) where e1 and 02 are some positive scalars. Assume that for some constant L > 0, for all w, E R we have [ (w) Vf (w)ll I L ||ue w|l, and that on 0, > 1/t co. Then
either f(w)) + -00 or else f(wx) converges to a finite value and lim V (((,)) =0. O.
Proof. We begin by establishing a standard descent inequality. Fix w, 2 E R" and define g f'w + {z) for & E [o, 1]. By the chain rule, g' (E) (z, V + fz)). Applying the fundamental theorem of calculus,
s(w + 2) - ((w) - ((1) - -(0)= L. ((()) dE S (z. +(( de
(z, vf(w)) + / (2, ff(w + vf(w)) de
< (2, Vf(w)) + / ||||| [VI + V v/(w))] de
S (z. Vf(w)) + |=|| / =E||=||dE
(z, (f())) + T. ||=113.
Now set 1/22. Using also condition 1 and 2 we can write
L
L 2 |lee| |l = f(we) + me (o Vf(ww))
f(w++1) S { + (nee,, Vf(w)))
2
L
S [(w)) - -naa -naillV/(uxx))
2 (V:)))
2 1/2
= =f(w))= T7 (a Lo3 1
(w:))
Since 7/c - 0, there exists a positive constant e such that for all t beyond some index F, E, f(w++1) S f(wx) - n/c | (wx)) This inequality implies that for / 2 [, the sequence f((wx) is monotonically non-increasing. Consequently, either f(w)) + -o0, in which case the proof is complete, or f(w,) converges to a finite limit. We now proceed under the latter.
To prove that lim4-400 fw() 0, suppose the contrary, namely that lim sup [ 0. Then there exists > 0 such that (w))) < e/2 for infinitely many t, while |V f(w)] > e for infinitely many t. Hence, we can extract an infinite subset of indices T such that for each E T. there exists i(t) > with
[ < c/2,
[Vf(wa)) >c. </2 sc, if t
Meanwhile, because
[V - [V/(())) <
[V - ( < I|| W144 we || s m Lc2 [f(w))),,
it follows that for all t E T that are sufficiently large so that Le2 < 1, we have < [wx)l: otherwise the condition e/2 < / (w]+)) would be contradicted. Without loss of generality, we assume that this relation holds for all E T.
We have for all E T, using the condition ||>=|| S c2/ (wx))] and the Lipschitz condition:
< [V - -[V] < [V/(w))) [w;)))
2
10-1
< Lllwan - will <
Σ 1
s(1)1
-(t)11
< Loz Σ
Thl [\(w))] < Loze
Σ
i=
1=
38

---

Published as a conference paper at ICLR 2026
which gives Σ ni 2 2Ley large T, and the relation || [w;)]l e/4 for i t, ,,..,, i(()) we can write
1. Now using ł, f(w++1) - f(w) n/ / for sufficiently
f(wat)) S f(ws) (9) =
Σ nh VtET.
Since f(re) converges to a finite value, the preceding relation implies that lim LET 2 0 < 2Log Therefore, the previous assumption is false, and we conclude that lim Vf(w)) 0. This concludes the proof.
-i(t)-1
D.3 PROOF OF THEOREM 1
Proof. The basic idea of this proof is to show that the configuration vector is upper bounded and that all three objective functions are p-Lipschitz. Then, since we assume the problem convex, we can directly obtain the final result via Lemma 4, First, according to the definition of オ, configuration ||=|] is clearly bounded by V12² p = B. Below, we discuss the Lipschitz properties of the functions.
Function Li's Lipschitz We can observe that fe is linear in the path weights and so in fe Extey Eper.,. Leep Da,t" Tp S Exsev Epep,. Eesp Dmnar Tp. So we have
4i (x())= Li (ntii
Se (T:(1) x1())
D Dinax || |x((1 T(51
max
Ce
Cmin
Therefore, L; is p-Lipschitz, with Ponn
Cyntas
Function L.'s Lipschitz From the definition of the maximum total flow, one should notice that C, is the sum of fp. Therefore, it suffices to show that fp is p-Lipschitz, in which case L, is simply Ep p-iisschitz. To focus on [[p (Ir()) - fp (r(1)) ||. we distinguish between two cases according
the value of min mine (m): = sb and make the following observation: S x.(>) S Cmax (x)))
(700)= 1. We have that
|1, (x()) So (TT) ||,0 (z1() 1b (x(1) xp (Tt11
-5
Without loss of generality, we assume that Sp (Tr(1) 2 Sp (1)) Then
112 (x)) -sp (T xx (1)() vs (z()) xpp (m1)) xp (z)( xxp (Tt(1))
Also note that (T)(T) Dz,t" Typ where Tp n. Therefore,
fr (x(0) (n() D Dmar |r(0)) - niii
Cmar Dmax 1 x(0
Cmin
39

---

Published as a conference paper at ICLR 2026
Φ (T(1)
1. Under the same assumption above, we additionally set a (T(i))
Gey
mey (+01))
8r
π
fp
=p
(8)
16 (1)
xp
T (5)
3 (T())
= w (0)
x (5)
π (1)
d ( )
+ v (
da T (8)
3
(aae) da. (x(1)
p
1
da (
T. x
4 (x(6)
(o0s))
x
os (x6))
( (200)
S 46
12 +
(7(60) T
CE2
ce2
(
- (os(
2 (8)
(5)
( d.-
5e (x01)
+ 2 (T(
z (2(3))
dx (T()))
Ges x (x((0)
Tes (=())
S s (
+ zp (1())
zt (1))7)
F Ce,
Ge2 T (x(())
Cmar
Ip (x()))
(x(o))
Zp (T)
Ip (m(0)
+
Tp (T))
Geş
Ge3
Cmax
+ x (1)
Tp (T))))
(ee())))) - F (x(s))
Gmin
where the second inequality derived using la + bl S |a| + bb). Recall that xe((x) Eesp xp((T) =
Eeep Da,t" Tp. and xy (z)(t) xp (Tr(1) s Dinar (+) x x010 ||. Therefore,
50 (x(1)) - fr (T(0)
Cmaz
D Dmaz
|r(( - x x + D Dmaa |T7( - n6
Omin
Cmaz
Donax
[ - x001 -000
Cmin
By combining two cases, can conclude that the function fp is p-Lipschitz, where p is at most As result, L. E., Epep,, fp Lipschitz, and its Lipschitz constant is at most
2.0mux Dimax
Gmin
Ee,t EpePar.
2.0max D
Conin
Function Le's Lipschitz The proof for Le follows essentially the same idea for Lt, as both focus on the Lipschitz property of fp. Hence, we can dircctly reuse the result established in the proof of L, for fp. And C. is also guaranteed to be p-Lipschitz, with p at most Epep,, 2.2mar Cmaz D
We have now established that all three objective functions L are p-Lipschitz and convex, and that || is bounded by B. Finally, by Lemma 4, we can write: For every e > O, if we run the algorithm on C
Ba for finite T (at least B²p steps with 1) = v pr then the output vector satisfies f(a) - f((T) Sc. which completes the proof.
D.4 PROOF OF LEMMA I
Proof. We consider the following expected policy gradient (Foerster et al., 2018) of PRAM:
== E, Σ A((8,a) a) Vo loggooaa sx) (5)) where A,(x,a) = R(s,a) a) - To(al 81) R(s,, (a_., d()).
40

---

Published as a conference paper at ICLR 2026
Therefore,
g= Σ p(s))T((a [s) Σ Σ To (a) (s))) R(s, (a=:: Vo log To s;)
Σ p(s)TT((a s) Σ R(s
ER(s, a) Vo log mo(a; ss).
Vanishing of 91-
For each agent i, the corresponding component (2) can be written
o = Σ p(s) Σ aa- s-1) Σ Talas ss))
Σ To st) (c.) Ve log To(a;
Σ (())_o(a 8 (,)) Σ To(as 81) Vo log moga (..))
s,a-;
where the inner term B, == Ea; Ro(ds) s;) R(s, (a-i; a()) is independent of az Since
Σ To(a; s;) Vo Vo mola s;) Σ VoTo(a s;) = Vo Tmo(a si)) Vo(1) = 0,
it follows that g = 0 for all i, and hence 91 = Ei 91 (1) = 0.
Analysis of 92- As discussed in 4, under our MCF setting (decoupling of rewards and states), the long-term return satisfies Q* = R(s, a). Thus, (1) can be rewritten as (s, a) Vg log Tp(a; s:)] which coincides with the standard policy gradient (i) Ex (s,
L2
formulation (Williams, 1992). Moreover, by Lemma 5, this admits the equivalent expression (r(s,a) Q" (s)
(8) E. dd ds) Ea
06
Suppose fw is a function approximator of Q", parameterized by w and locally optimal. Without loss of generality, assume
0f.. (s.a) 8w
Under this construction, Lemma 6 implies [(.,0] 88
Ōn(s.a)
28 = Σ. (s) 36
@n(s.a) fs,
80
We may define the update direction ek = E, d (s) E. 05 (s.a) sures (s, a), since the ((R)_ update follows the policy gradient direction. The boundedness of Vi[((s, a) ensures, via the mean value theorem, that VeR(s, a) is Lipschitz. Furthermore, 2n satisfies the required condi- tions: //Vo R(s, a))2 < -oc R(s, a) d and || < ||V R(s, a)ll, since 2) is an unbiased estimator of the policy gradient. Together with the step-size requirements and bounded function, these are precisely the conditions needed to invoke Lemma 7, which guarantees convergence to local optimum, i.c., lim4 92 lim E, Vo(e) R(s, a) =
Finally, we have limg- g = 0 + limtg-ro //2 O, which completes the proof.
D.5 PROOF OF THBBREM 2
Proof. Under Assumptions and 3, we first analyze the dynamics induced by gradient descent. Consider the reference linear model y(x) Wx and fine-tuning dataset consisting of input embeddings x; and corresponding decision embeddings y/- The training objective is the squared-error loss L(W)- 1
2N Σ [[: Wx: 22 One step of gradient descent with learning rate updates the weights as AW = -1 WwWWWW
2 >."(Wa,, (Wa,, w.; y:) Substituting into the loss gives
N
N
L(W + AW) 2N Σ
[[( + AW)x, De 2
Σ
Wx, - (y, - Ay:))l
2N
41

---

Published as a conference paper at ICLR 2026
Now consider the linear self-attention update on the token Cj.
ej ej ATT(feI,......
(ee(ee,,..,,,)))) eNN)) ej + Σ PAVAKK mhg-
The key idea is to construct WK, Wo, Wv and the projection F such that single Transformer
update on token ej replicates the gradient-descent dynamics: eg (2g, + (0,=AWa,) (z Mj Ay;). Specifically, consider the block-structured weight matrices for one attention head:
WK WQq = (os :). Wv ( Wo -1n)
0
where La and ly are identity matrices of size N₂ and Ny, and Wo E RN is the reference model. We also set P #1, where I is the identity matrix of dimension N₂ Ny. With this construction (Ahn et al., 2023).
Arr(fet....,eN) 1EEmR Ph Σ th kh. Σ PhWn. Een, cne &
and we obtain the following update dynamics:
I2
Iz
7
I2 yk
L
4
Lx
Z y5
x
Ui
U5
Wo
-Iy
0
y4
0
N
F2 0
Vi
Wor,
0
Izj yj
0 AW.,)
Thus, each token ej (xjj including the query token eN+1 Ptsst (Frest: Wo.eestt) follows exactly the gradient-descen update. By stacking multiple attention blocks and heads, we can therefore simulate multiple steps of gradient descent, completing the proof.
42

---

Published as a conference paper at ICLR 2026
Mac Line Uhage
To08 730
Mox
Narther Layers
Narber Lepois
Marbbrr MLMI's Lagars
Narbbr Lagors
(a) Abilene
(b) GÉANT
Figure 20: (normalized) Performance comparison with different number layers in the ML.M.
E ADDITIONAL EMPIRICAL RESULTS
In this section, we present supplementary experiments to further explore the effectiveness of PRAM. We first analysis the impact of multimodal language model's layers and backbones. Then, we investigate the performance of PRAM with different synthetic data distribution.
E. IMPACT OF MLM'S BACKBONE MODEL
During our main experiments, we kept the LM backbone fixed to sim- plify presentation and ensure fair com- parison. In this subsection, how-
Max: Link Unage (4)
Abllene
GÉANT
UsCarrier
Cogenico
ever, besides E Qwen2.5-VI--7B- Instruct, further report PRAM'S performance with alternative pre- trained backbones of different scales,
1.0
Total Flow (1)
including the larger ( Llama-3.2-
LIB-Vision-Instruct (Meta, 2024), and the smaller
Qwen2.5-VI--BB-
Avg. Concur. Fow (T)
Instruct (Bai et al., 2025). As de- scribed in our experimental setup, all MLMs are truncated
io
layers. No-
tably, there exists substantial archi- tectural difference between Llama and Qwen: while Qwen encodes visual features as part of the input "prompt," Llama integrates them across multiple layers via cross-atted cross-ateention. To maintain comparability, we truncate only those layers of Llama-3.2-11 Des-L----ii--nnnnnenee language model that do not involve cross- cross-attention.
0.c
O4e12.5M--8
Figure 19: (normalized) Performance comparison with dif- ferent MLMM backbone models.
The results, presented in Figure 19, show that most MLMs perform well on the target MCF problems. Qwen2.5-VL-77 Inntrrutttt consistently performs worse than Qwen2.5- VL-3B-IIntrree which aligns with our expectations, as more parameters often lead to stronger understanding and reasoning capabilities. But interestingly, despite having fewer parameters, 10wen2.5-VI-7BB-IG
demon- strates stronger planning ability than the larger Llama-3. 3.2-11BB Vission-Innttruct We attribute this to Qwen's training strategy, which emphasizes numerical reasoning and step-by-step problem solving, whereas Llama, although powerful in language generation, less specialized in mathematical do- mains. This suggests that model scale alone does guarantee superior performance the MCF problem. Therefore, we choose Qween2.5-VL--B-Insstnn as the default backbone for PRAM.
E.2 IMPACT NUMBER MLM'S LAYERS
In Figure 20, we show how PRAM'S link utilization and throughput (total flow) change with the number of Transformer layers across different datasets. The results clearly indicate that performance improves the number of layers increases, with the effect being particularly pronounced on the Abilene dataset.
43

---

zi Published as a conference paper at ICLR 2026
0.00
0.05
0.60
0.80
Total Flow
0.75
0.70
0.6
Total
0.70
ARB84 Meen
Sexarnal Nate Trend
0.98
Seasonal
0.0
Moving
0.80
Koongguucrage
POP (pred)
LP (pred)
LP4op (pred)
(pred)
(prod)
(pred)
(a) Abilene
(b) GÉANT
Figure 21: (normalized) Performance comparison with different prediction methods.
This finding is consistent with the sharp deterioration observed in our main experiments when the backbone is entirely removed (i.e., # of Transformer layers = 0). We attribute this to the heterogeneous allocation patterns in Abilene, which demand stronger representation capacity (i.e., deeper models) to capture effectively. Nonetheless, the performance gain diminishes beyond 8 layers. To strike balance between fine-tuning efficiency and performance, we adopt 8 layers as the default setting in main experiments.
E.3 BASELINE RESULTS W.R.T. PREDICTION ALGORITHMS
Our evaluation in $ focused demand- prediction-based schemes using simple moving average. We now compare the moving average against other prediction algorithms on real-world datasets (Abilene and G'EANT). We restrict our compari- son to non-parametric methods, since para- metric methods such as regression incur higher computational cost and suffer from significant eиrors due to denormalization of DNN outputs. Specifically, we consider the following prediction methods: ! Mean Value: predicts the average of the entire his-
10
Moan Value
Mean
Awerage
Normalized MAE
tory; 2 Seasonal Naive: repeats the last observed season into the future; 3 Linear Trend: performs linear extrapolation based on the last two observations; and ④ AutoRegressive Integrated Moving Average (ARIMA, Shumway Stoffer (2017)): widely used statistical model for time series forecasting. Each method predicts the next demand for every source-destinate pair using only the 12 most recent observed demands of that pair.
Abilene
GEANT
Figure 22: Accuracy of different prediction methods.
Figure 22 presents the accuracy of the different predictors, measured by the normalized mean absolute error (NMAE), i.c., & ΣK [z-=d,l As shown, ARIMA and the moving average achieve the lowest errors average across both datasets, although ARIMA requires more time collect sufficient information. In contrast, the linear trend exhibits high instability, and the remaining predictors clearly less accurate. Figure 21 further illustrates the impact of different predictors multi- commodity flow (MCF) allocation, quantified by total flow relative to an optimal LP solver. The trends mirror the accuracy results: ARIMA performs best, followed by the moving average. while linear trend underperforms compared to mean value despite its seemingly aggressive predictions. Overall, considering all evaluation metrics and computational cost, we adopt the moving average as the default predictor for our baseline methods.
E.4 RESULTS W.R.T. SYNTHETIC DATA DISTRIBUTIONS
In this subsection, we provide a supplementary study to our evaluation in $ 5.2, where we synthesize demand matrices for large-scale topologies using broader set of distributions, including the Gravity,
44

---

Published as a conference paper at ICLR 2026
Average Normalizzdd Results using Gravity Model
Flow Total
Un
grod)
jrofi
(a) Gravity Model
Average Normalized Results using Poisson Model
Likk
0.5
Tote
Ave. Contur How
Colt
UoCarreer
(b) Poisson Model
Average Normalized Results using Bimodal Model
Usage
0.
1.0
Avg.
(c) Bimodal Model
Figure 23: Performance comparison on large-scale topologies with different data distribution.
45

---

Published as a conference paper at ICLR 2026
Example of Visual Understanding and Responses
You given (directed) network topology diagram the attached image. The network called
GEANT contains 22 nodes, and the image shows the sub-topology capturing all communication paths from the node (marked red) to every other node.
Your task:
Extract every link (edge) visible in the sub-topology.
2. For each link, record:
The two endpoint nodes (e.g.. (A, B) denotes link from A to B):
The link capacity shown in the diagram (any numerical or annotated capacity value).
3. Do not invent links capacities. Only report information that is present in the image.
4. Provide the final output in the following format:
Links: (A, B): capacity
Additional Notes:
Links:
(9. 0): capacity 100,0
(8, 19): capacity 24.0
19): capacity 24.0
(0, 2): capacity = 24.0
(0, 4) capacity 100.0
(0, 19): capacity 24.0
(2, 6): capacity 100.0
100000
(2, 21): capacity 24.0 x
4): capacity 100.0
X
(4, 14): capacity 100.0
(4, 10): capacity 24.0 24.00
(4, 12): capacity 100.0
(4, 21): capacity 100.0 10000
(3, 16): capacity 24.0
4) capacity 100.0
(18, capacity 100.0
(10, 21): capacity 24.0
(12, 7): capacity 24.0
(12, 11): capacity 1.55
(2, 12): capacity 100.0
(14, 11): capacity 1.55
(14, 11): capacity 1.55
(5, 17): capacity 24.0
(6, 5): capacity 100.0
1): capacity 24.0
(4, 6): capacity 100.0
(6, 13): capacity 1.55
(1, 13): capacity 1.55
Additional Notes:
(1, 14): capacity= 24.0
(21, 17): capacity 24.0
No link has a missing capacity label; visible links have explicit numerical cap- acities shown. Capacities are labeled dir- (18, 21): capacity 100,0
(21, 6): capacity 100.0
(21, 5): capacity 24.0
(21, 10): capacity 24.0
(20, 3): capacity 24.0
ectly on the links and are
Table Pretrained MLM's visual comprehension example on sub-topology with overlapping ele- ments. The task involves extracting every visible link (edge) within the specified sub-topology and recording the two endpoint nodes (A, B) and the associated numerical link capacity. The (V) and (x) indicate the model's success failure in identifying the corresponding link and its capacity value, demonstrating the visual comprehension level of technical diagrams.
Poisson, and Bimodal models (see Appendix B.2.2 for details). Our results are in Figure 23. Note that the results of the gravity model in Figure 23(a) have already been reported in the main text, we refrain from elaborating on it here benchmark. Since the std is very small for different samples (see Figure 9(b) (right)), we omit the drawing of error bars here.
First, consider the Poisson model in Figure 23(b), where we observe pronounced performance fluctuations across methods as the topology changes. Particularly on GtsCe topology, POP exhibits poor maximum flow preservation. Somewhat counterintuitively almost all LP-based methods perform better link utilization prediction than the exact method. We attribute this to the randomness of the Poisson distribution, which selects aggregation points independently each step, leading to increasingly chaotic commodity flows. Among the learning- based methods, DRL consistently underperforms, while PRAM demonstrates robustness with only negligible fluctuations.
Next, as shown in Figure 23(c), our first observation under the Bimodal model that all methods achieve near-optimal performance terms of concurrent flow. This is expected, since the synthetic
46

---

Published as a conference paper at ICLR 2026
traffic is highly sparse: fewer than 10% of the flows are relatively large, while the rest close to zero, making average parallel flow optimization straightforward. However, prediction errors become more pronounced due to the increased magnitude of the dominant flows. While PRAM is largely resilient to such errors, it performs slightly sub-optimally on the maximum flow objective, ranking just behind L.P and L.Pttpp
To sum up, across diverse demand distributions, PRAM consistently demonstrates robustness and adaptability. Unlike DRL, which struggles with instability, LP-based methods, whose performance is sensitive to distributional assumptions, PRAM maintains stable results across objectives and topologies. This suggests that PRAM not only generalizes beyond the traffic models used for training but also avoids overfitting to specific flow structures, thereby providing a more reliable solution in practical large-scale settings.
E.5 EXAMPLES OF MLM'S VISUAL UNDERSTANDING
To ensure the used off-the- MLM's visual understanding capabilities, we conduct some specially designed question-ans answering small experiments with Qwen in the subsection. Our pipeline is very simple and consists mainly of two stages: a) To simulate the iterative process during fine-tuning, meaning the MLM should understand the overall topology information, we first input all subgraphs to the model once; b) We then perform the formal question-answering (QA) experiment, instructing to provide as much of the observed image information as possible.
As illustrated in Table 3, start by challenging test case for evaluating the ability of MLMs to accurately parse and extract structured information in the presence of visual occlusion. The specified GÉANT sub-topology (communication paths from node 9, marked in red) contains several indistinguishable elements resulting from the intersection and overlap of edges and nodes (e.g.. 2 + 12 and 2 > 6). For view and report links, MLMs can achieve good performance with an accuracy of nearly 90%. A important finding here is that the majority of erroneous answers are not the occluded ones. This can be attributed to a) the global information previously presented to the multimodal model, which, some extent, provides the MLM with overlapping structural information from multiple perspectives. Next, Table presents results on two topologies of different sizes: Abilene and UsCarrier. Note that since the MI. refused to answer based on observations from the ultra-large topology, we segmented the corresponding diagram into multiple smaller images for recognition. The accuracy on Abilene is almost zero error, and the results on UsCarrier did not degrade significantly due to node density. Finally, we conduct QA on GÉANT with different image quality. In Table 5, image resolutions of 512 x 512 and 128 128 are used and compared. The performance reduced by ~00; however, we consider reasonable to appropriately compress the resolution to accelerate model inference.
is noteworthy that the MLM achieves the above performance without fine-tuning, showcasing its powerful and general vision understanding capabilities nowadays. In real-world scenarios, such as cloud environments or data centers, we anticipate having sufficient time and resources to adapt the MLM (via our lightweight method), allowing it to further correct and refine its information extraction from these network sub-images.
E.6 RESULTS W.R.T. SUBGRAPH PROCESSING SCHEMES
In this part, conduct an investigation into various methodologies for representing subgraph as content that can be understood by MLMs. Beyond the Vision-based scheme used by PRAM, we also consider two variants, namely the Text-based scheme and the GNN-based scheme. Following the best practice of Fatemi et al. (2024), baxt- scheme encodes graphs as text in an Incident style. For example, we can add "Here is graph among nodes 0~8. In this graph: Node 0 is connected to nodes 1, 2. Node 1 is connected to nodes Node
0, 2. Node is connected to nodes 0, 1, 3, 4, 5, 7 is connected to nodes 3, 7." into the original prompt for encoding any subgraphs. Regrading GNN-based scheme, although this contradicts our intention to avoid manual design, separately design encoder, which is 3-layer Graph Transformer architecture using Py PoTch Geometric library, for processing graph-structured inputs, where the node features consist of each node's 's degree and the sum of the capacities of its adjacent links. Analogous to the aforementioned
47

---

Published as a conference paper at ICLR 2026
Example of Visual Understanding and Responses
You are given (directed) network topology diagram as the attached image. The network called
Abilene (or UsCarrier) contains 12 (or 197) nodes, the image shows the sub-topology capturing all communication paths from the node (or 1) (marked in red) to every other node.
Your task:
Extract every link (edge. visible the sub-topology.
2. For each link, record:
TTee two endpoint nodes (e.g. (A, B) denotes link from to B):
The link capacity shown in the diagram (any numerical annotated capacity value).
3. Do not invent links capacities. Only report information that is present in the image.
4. Provide the final output in the following format:
Links: (A, B): capacity
\n Additional Notes:
Links:
(0, 1): capacity 992.0
(1, 4) capacity: 992.0
(1, 5): capacity 248.0
(1, 11): capacity 992.0
(4, 7): capacity 992.0
992.0
(5, 2): capacity 992.0
6): capacity 992.0
(6, 3}: capacity 992.0
(7, 9): capacity 992.0
(2, 8): capacity 992.0
(9, 3): capacity = 992.0
(9, 10): capacity = 992.0
10): capacity 992.0 992.0
8): capacity 992.0
Additional Notes:
All links are directed, indicated by arrows
edges: All capacities numerical values, likely repre-
senting bandwidth in Mbps or similar units.
Links:
(1, 103): capacity no capacity shown
(1, 85): capacity no capacity shown
(1, 86): capacity capacity shown
(112, 115): capacity no capacity shown
(77, 135): capacity no capacity shown
(53, 52): capacity capacity shown
(21, 20): capacity no capacity shown
(69, 108): capacity capacity shown
Additional Notes:
Due to visual complexity and overlapping edges in the topology figure, some links may not be fully extract- able from the image alone. The interpretation is thus limited to structures
Table 4: Pretrained MIM's visual comprehension example sub-topology with different scale (density). The task involves extracting every visible link (edge) within the specified sub-topology and recording the two endpoint nodes (A, B) and the associated numerical link capacity. The (v)
and (x) indicate the model's success or failure in identifying the corresponding link and its capacity value, demonstrating the visual comprehension level of technical diagrams.
48

---

Published as a conference paper at ICLR 2026
Example of Visual Understanding and Responses
You are given a (directed) network topology diagram as attached image. network called
GÉANT contains 22 nodes, and the image shows the sub-topology capturing all communication paths from the node O (marked in red) to every other node.
Your task:
1. Extract every link (edge) visible in the sub-topology.
2. For each link, record:
The two endpoint nodes (e.g.. (A, denotes link from A B):
The link capacity shown in the diagram (any numerical or annotated capacity value).
3. Do not invent links capacities. Only report information that is present in the image.
4. Provide the final output in the following format:
Links: (A, B): capacity
(C, D): capacity=
Links:
(0, 4) capacity 10.0
(0. 9): capacity 10.0
(0, 15): capacity 2,4
(0, 19): capacity 2,4
18): capocity = 10.0
(6, 4) capacity 10.0
(4, 10): capacity 2.4
(4, 12): capacity 10.0
(4, 14): capacity 10.0 10.0
(21, 10): capacity = 2.4
(2, 6): capacity 10.0
18): capacity 10,0
(2. 12): capacity=100 10.0
(6, 1): capacity 2.4
(6, 5): capacity = 10.0 10.0
(6, 13): capacity= 0.155
(6, 21): capacity 10.0 10.0
(1, 13): copacity 0.155
(5, 12): capacity 10.0 10.0
(5, 17): capacity = 2.4
(14, 11): capacity 0.155
(14, 21): capacity 10.0
(12, 7): capacity 2.4
512
512
Links:
(0, 4) capacity 10.0
(0, 9): capacity 10.0
(0, 15): capacity 2.4 2.4
(0, 19): capacity 2.4
2}: capacity 10.0
4) capacity 10.0
(4. 10): capacity 2.4
(4. 12): capacity 10.0
(4, 14): 1opacity=10.0
(21, 10): capacity =
(2, 6): capacity 10.0
(2, 18): capacity 10.0
(2, 12): capacity= 10.0
(6, 1): capacity = 2.4
5): capacity 10.0
13): capacity 0.155
(6, 21): capacity 10.0
(1, 13): copacity 0,155
(5, 14): capacity 10.0 10.00
(5. 17): capacity 2.4
(14, 12): capacity 2.4 2.4
(14, 21): capacity 10.0
(12, capacity 2.4
128
128
Table Pretrained MLM's visual comprehension example on sub-topology with different resolution. The task involves extracting every visible link (edge) within the specified sub-topology and recording the two endpoint nodes (A, B) and the associated numerical link capacity. The (v) and (x) indicate the model's 's success failure in identifying the corresponding link and its capacity value, demon- strating the visual comprehension level of technical diagrams.
49

---

Published as a conference paper at ICLR 2026
View sovd Siner
DVeilssed Riterea
Wore wwweeeeatiin Adreen
ONN
More Lime Uraqge
1.0
Tosel
Time
Time
Time
Time
(a) Abilene
(b) GÉANT
Figure 24: (normalized) Performance comparison with different subgraph processing scheme.
Len: 50-149
150-249
Len: 250-349
Len: 350-449
Len: 450-549
0.22
0.20
Dansity
0.08
0.06
0.00
0.52
105
200
300
100
200
320
1.50
200
300
100
200
300
1.50
203
100
Latency (ma)
Labency (ms)
Labency (ms)
Latency (ms)
Latency (ma)
Figure 25: Inference latency distribution vs. input token length of language models.
Qwen architecture, we concatenate the GNN output features with the textual embeddings to form the input for the backbone model.
As depicted in Figure 24, the adapted MLMs outperform the learning- baselines presented in the main text. This suggests that the reasoning capabilities of MLMs indeed contribute solving the multi-commmdity flow problem. However, it can be seen that there is clear ranking among the three schemes for maximizing total flow on GÉANT, where the Vision-based scheme consistently yield improvement over other two schemes. Although the performance of textual encoding exhibits high variance, ranks second overall, surpassing the average results of GNN encoding, due to better understanding of the modality. It is also noteworthy that the topology scales, the increasing complexity of the description leads to rapid growth in prompt length. To investigate the inference overhead of Text-based scheme associated with this length, specifically conducted simulation experiments. The sampled latency results are presented in Figure 25, We observe a pronounced linear increase in inference latency with respect the input token length. This implies that, in practice, on larger and more complex topologies, Text-based schemes are likely to suffer a precipitous drop in efficiency owing to their significantly inferior compression ratio compared visual approaches.
50

---

Published as a conference paper at ICLR 2026
F LIMITATIONS & FUTURE WORKS
Several limitations of PRAM warrant consideration. First, its effectiveness relies heavily on the underlying pretrained MLMs, which were originally developed for natural language and image processing rather than MCF problem solving. When these pretrained models face resource constraints, as discussed in $ 6, their ability capture global patterns may be impaired. Second, PRAM currently adopts non autoregressive generation paradigm for solving general MCF problems. While recent studies have shown promising results with autoregressive formulations for temporal sequences (Liu et al., 2024), incorporating such techniques into PRAM represents valuable avenue for future exploration. Third, although frozen ML are employed to improve computational efficiency, the training process remains time-consuming, which may limit applicability in time-sensitive settings. In addition, processing identical sub-images can still incur significant memory overhead.
Despite these challenges, PRAM introduces an innovative foundation model architecture specifically tailored for MCF problem solving, delivering gains in both efficiency and solution quality. This work lays the groundwork for future advances in scalability and resource optimization. Moving forward, we plan to explore more sophisticated fine-tuning strategies to further enhance performance while reducing computational demands, drawing recent progress in parameter-efficien adaptation as well as memory storage and retrieval mechanisms.
G USAGE OF LLMS
We acknowledge the use of LLMs as a general-purposs assist tool during the preparation of this paper. The LLM used to help with refinement (e.g., improving clarity, grammar, and style in $ and $ 4) and with programming assistance (e.g., generating plotting scripts and formatting code for 5). All ideas, research methodology, analyses, and conclusions are entirely those of the authors.
51


Reviews:
[{"review_id": "fXfMwH6n1w", "reviewer": "Reviewer_j7pq", "summary": "The paper proposes PRAM, a novel framework that applies multimodal large language models (MLMs) to solve the multi-commodity flow (MCF) problem. Instead of handling large-scale MCFs as a single optimization, PRAM decomposes the process into a structured Divide–Harmonize–Conquer pipeline.\n\n1. Divide:  \n   The global network is partitioned into node-based subproblems. Each subproblem is encoded as a multimodal input, with network topology represented as images and demand profiles represented as text.\n\n2. Harmonize:  \n   A collection of MLM-based agents is coordinated via a lightweight multi-agent reinforcement learning framework, enhanced by LoRA (Low-Rank Adaptation) and learned context embeddings for efficient cross-agent communication.\n\n3. Conquer:  \n   The framework achieves global convergence through the implicit gradient-descent-in-context property of MLMs, allowing distributed subproblem solutions to align toward an optimal global flow.\n\n\nEmpirically, PRAM demonstrates near-optimal allocations across ten real and synthetic network topologies, often outperforming classical LP-based solvers by 1–2 orders of magnitude in runtime while maintaining <8% optimality gap. Theoretically, the paper establishes convergence guarantees under convexity assumptions and provides an interpretability analysis of the learned context embeddings.", "questions": "1. How is the visual encoding of topology constructed, and how sensitive are results to the layout or resolution of the generated subgraphs?\n\n2. Would representing the topology as a matrix or sequence (instead of an image) yield comparable results?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. Using multimodal LLMs as distributed solvers for MCF is genuinely new. Prior work on neural optimization largely relied on GNNs or RL; this paper reframes the task as multimodal reasoning over partitioned problems, which feels like a meaningful step forward.\n\n2. The paper’s theoretical discussion—particularly the connection between convexity in MCF and the ability of MLMs to simulate gradient descent is well argued and gives the approach more credibility than most ML-for-optimization papers.\n\n3. The experiments are comprehensive, covering several datasets and objectives. PRAM consistently performs close to LP while running 10–100× faster. The robustness tests (link failures, demand fluctuations) also support the generalization claim.\n\n4. The choice to use LoRA and shared context embeddings keeps the model efficient. The MARL setup is simple but effective for ensuring coordination between agents.", "weaknesses": "1. The multimodal input design (topology as image + demand as text) is acknowledged to introduce layout sensitivity and encoding bias.\n2. Although faster than LP in inference, fine-tuning large MLMs still requires considerable resources.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "y5Y931SQ5X", "reviewer": "Reviewer_X9xi", "summary": "The paper showcases an agentic AI application leveraging multimodal large language models (MLLMs) to solve the multi-commodity flow (MCF) problem through a multi-agent Reinforcement Learning (RL) framework. The method aims to resolve the scalability bottleneck of the classic solver by first dividing the network into subnetworks, then applying MLLM agents to solve each one, and finally ensuring global consistency through the harmonization of the subproblems.", "questions": "1. **Convergence**. My understanding of Theorem 1, Theorem 2, and Lemma 1 is that they only guarantee convergence for PRAM to locally optimal agents with a limited number of training steps. How can we guarantee that PRAM can strictly learn a global optimal set of agents?\n2. **MARL Adaptation**. The MARL algorithm uses counterfactual reasoning, asking how the global objective would change if one agent acted differently. Does the computational cost of this \"counterfactual baseline\" scale linearly with the number of agents (i.e., the number of nodes )? If so, could this harmonization step become a new bottleneck for extremely large topologies, even if the problem is partitioned?\n3. **Comparison to LP Solvers**. The paper notes that PRAM's results for minimizing link usage were \"even better than those of LPs\". Based on my understanding, since LP solvers are guaranteed to find a near-optimal solution given a specific demand matrix, how is this possible? Does this imply the LP solver was using predicted demands, while PRAM (using historical data) generalized better to the actual future demands used for evaluation?\n4. **Role of _Reasoning_**. The paper claims the MLM \"_...learns to perform gradient descent in context..._\". Is this an emergent, implicit behavior, or is the model explicitly structured to perform GD steps? The proof for Theorem 2 constructs specific attention weights to demonstrate that this is possible, but does the model actually learn this specific structure during fine-tuning?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Novelty.** To the best of my knowledge, this paper is among the first to leverage the reasoning power of modern Multimodal Language Models (MLMs) for a classic combinatorial optimization problem (MCF). It bridges the gap between large-scale AI models and network optimization.\n2. **Significant Performance (Speed and Scalability).** Based on the results, PRAM is significantly faster on large-scale topologies. The ablation study confirms that the partitioning is critical to this scalability.\n3. **Near-Optimal Allocation Quality.** Unlike previous ML methods (like DRL) that sacrifice quality for speed , PRAM achieves near-optimal performance, often ranking just behind an LP solver with perfect future knowledge. In some cases (minimizing link usage), it even surpassed the LP solver's results.\n4. **Strong Generalization and Robustness.** PRAM demonstrates a key advantage of ML-based approaches: it generalizes well to unforeseen conditions. It shows minimal performance degradation (<10-16%) during simulated link failures and demand fluctuations, scenarios where traditional prediction-based solvers struggle.\n5. **Solid Theoretical Foundation.** The paper provides a convincing theoretical justification for why PRAM works. It connects the problem's structure (convexity of MCF objectives, Theorem 1) to the model's learned capability (MLMs can simulate gradient descent, Theorem 2).\n6. **Thorough and Convincing Evaluation.** The experimental validation is comprehensive. It leverages multiple real-world and large-scale synthetic datasets , tests three different MCF objectives , and includes a detailed ablation study that validates the importance of every major component (partitioning, the MLM backbone, LoRA, context, and MARL).", "weaknesses": "1. **Cost in Training**. From my understanding, the superior scalability of PRAM is achieved by truncation of backbone layers and parameter-efficient adaptation. This can potentially be a problem during deployment.\n2. **Memory overhead**. Input to the model are images of the subnetworks, which are sensitive to the partition parameters. I wonder if this can lead to significant memory overhead, especially when different subproblems (nodes) have similar network views that are processed redundantly.\n3. **Dependency on Pretrained Backbone**. The success of PRAM is \"heavily\" dependent on the quality of the underlying pretrained MLM. I wonder how performance varies with different backbones.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ShQosrRxM9", "reviewer": "Reviewer_QZnd", "summary": "This paper presents PRAM a reinforcement pipeline architecture to solve\nmulti-commodity flow (MCF). PRAM works by decomposing MCF problems, compute the\nrepresentations of subproblems and recombine the subproblem representations to\nreturn a complete solution. The context-free representation computation is\nperformed by a vision encoder and tokenizer (enriched with a learned global\ncontext). Then graph and text vectors are passed to a multimodal language\nmodel to combine modalities and compute MCF subproblem representations. The\nglobal recombination is performed by multiagent reinforcement learning using\ncounterfactual policy gradient.\n\nCommunication between subproblems is implemented through LoRA on the multimodal\nlanguage model, and through the global context vectors which are the query of a\ncross-attention mechanism.\n\nExperimentally, the model shows strong results on real-world datasets and public\ntopologies, outperforming previous reinforcement learning approaches.\nInterestingly, this model exhibits robustness as it is able to solve\nproblems with link failures and flow bursts with performance degradation under\n10%.\n\n\nThe paper is very well written, and the method is novel (to the best of my\nknowledge). The reason for using an off-the-shelf LM is clearly motivated, and\nthe adaptation mechanism is thoroughly described (although quite convoluted) and\njustified in ablation studies.", "questions": "1. the use of a MLM is justified, but the model w/o MLM process the input in a\n  puzzling way. I can understand how the topology is processed by a GNN, but I\n  don't understand how the textual descriptions are handled by a feedforward\n  layer (size may change for instance). Can you elaborate? Is there a global\n  context vector in this case?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. Well written and clear\n2. A novel method based on decomposition\n3. Strong empirical results, with intersting robust predictions", "weaknesses": "- the neural architecture is quite complex", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Si5E5Shd9w", "reviewer": "Reviewer_Zyiq", "summary": "This paper proposed a LLM-driven solver for the multi-commondity flow problem. The problem is firstly decomposed as $|\\mathcal{V}|$ sub-problems, then the sub-problems are transformed into images, and a MLM is prompted to solve these sub-problems. A multi-agent reinforcement learning algorithm with counterfactual objective is applied to fine-tune the MLM, so that the sub-problems can be solved in a cooperative way that maximize the global objective. Experimental results show that its performance is comparable to traditional baselines.", "questions": "See the questions in \"weaknesses\".", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- This paper shows the potential of multimodal language models (MLMs) on the multi-commodity flow problem\n- Proper techniques applied to the MLM to improve the efficiency/quality, including LoRA, global context, and the reprogramming technique\n- Successful application of multi-agent reinforcement learning on MLMs with a counterfactual objective\n- Good ablation studies\n- Sufficient details in the appendix", "weaknesses": "1. The main claim of the paper, the necessity of MLMs, is not very sufficiently supported by empirical results. For example:\n- The paper mentioned that \"(MLMs) exhibit emergent abilities that were not explicitly programmed in to them during pre-training on massive data, such as mathematical problem reasoning and generalization to unforeseen conditions\" (line 66). I generally agree with this. However, there is no empirical result shown in this paper that MLMs generalize better than other DNNs on unforeseen conditions. In Figure 8, no other ML-based baselines are presented.\n- The paper mentioned that \"(MLMs) mitigate the costs of retraining and handcrafting specialized DNNs for complex inputs\" (line 69). However, even with LoRA, retraining an MLM is still typically considered more costly than traditional DNNs (especially considering the GPU memory requirements). I generally agree that MLMs can be a more \"universal\" solution than specialized DNNs, which would be easier to adapt between different type of optimization problems (ideally only with a change of the prompt) while other approaches may require a major change. However, this paper is scoped only on the MCF problem with very specific designs (e.g., the decomposion approach). In such a way, this advantage is not very relavant to this paper.\n\n2. The only ML-related baseline approach (Valadarsky et al., 2017) seems to be very old. While some more recent approaches like GDDR (Hope & Yoneki, 2021) and MARL-GNN (Bernárdez et al., 2021) are mentioned in the paper, they are not included in the baselines. \n\n3. Some claims in the paper are not very clear to me, including:\n- In the abstract, the paper says \"dividing the original problem into local subproblems via an MLM-powered agent\" (line 20). But if I understand correctly, the division is not done by an MLM agent but rather \"rule-based\" (\"treating commodities from the same source as one subset\", line 157).\n- The convexity of the problem is illustrated in a case study and described as an assumption (and all the proof rely on this assumption). Given the closed-form formulation of the problem (appendix B.1), can we make it clear whether it is convex or not?\n- The proof is based on the fact that \"gradient descent can effectively solve the problem\" (line 253). Unfortunately, gradient descent cannot effectively solve the problem. Gradient descent typically works for unconstrained problems or soft-constrained problems (so that the constraints can be transformed to a penalty term in the objective). However, this is a hard-constrained optimization problem, in which all the constraints in equation (1,2,3) should never be violated. In such a way, specific constrained optimization approaches should be applied (typically with KKT condition and lagrange multiplier). Or, empirically, if gradient descent is truely an effective choice, why not solve the problem simply with gradient descent?\n\n4. Using bitmap images to represent graphs can be pretty lossy, especially when the graph is complicated. For example, in the examples provided in the paper, some edge capacities cannot be clearly identified due to heavily crossed links (e.g., row 3, column 1 of Figure 15, the capacity between node 2 and 12 cannot be identified)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)