Preference-Driven Multi-Objective Combinatorial Optimization with Conditional Computation
Authors: Mingfeng Fan
Jianan Zhou
Yifeng Zhang
Yaoxin Wu
Jinbiao Chen
Guillaume Adrien Sartoretti
Venue: NeurIPS 2025 poster ()
Keywords: Multi-objective combinatorial optimization; preference optimization; conditional computation
Primary area: deep_learning

Abstract: Recent deep reinforcement learning methods have achieved remarkable success in solving multi-objective combinatorial optimization problems (MOCOPs) by decomposing them into multiple subproblems, each associated with a specific weight vector. However, these methods typically treat all subproblems equally and solve them using a single model, hindering the effective exploration of the solution space and thus leading to suboptimal performance. To overcome the limitation, we propose POCCO, a novel plug-and-play framework that enables adaptive selection of model structures for subproblems, which are subsequently optimized based on preference signals rather than explicit reward values. Specifically, we design a conditional computation block that routes subproblems to specialized neural architectures. Moreover, we propose a preference-driven optimization algorithm that learns pairwise preferences between winning and losing solutions. We evaluate the efficacy and versatility of POCCO by applying it to two state-of-the-art neural methods for MOCOPs. Experimental results across four classic MOCOP benchmarks demonstrate its significant superiority and strong generalization.

Preference-Driven Multi-Objective Combinatorial Optimization with Conditional Computation
Mingfeng Fan
Jianan Zhou"
National University of Singapore ming. fancnus. edu. sg
Nanyang Technological University j jianan004@e. ntu. edu.sg
Yifeng Zhang
Yaoxin Wu
National University of Singapore y yifengSu. nus. edu
Eindhoven University of Technology y. wu28tue. nl
Jinbiao Chen
Guillaume Adrien Sartoretti National University of Singapore gui Haalllamm..or sartorett. @nus edu
Sun Yat-sen University chenjb696mai12.. sysu. sysu.ddu.com edu. cn
Abstract
Recent deep reinforcement learning methods have achieved remarkable success in solving multi-objective combinatorial optimization problems (MOCOPs) by decomposing them into multiple subproblems, each associated with a specific weight vector. However, these methods typically treat all subproblems equally and solve them using a single model, hindering the effective exploration of the solution space and thus leading suboptimal performance. To overcome the limitation, we propose POCCO, novel plug-and-play framework that enables adaptive selection of model structures for subproblems, which are subsequently optimized based on preference signals rather than explicit reward values. Specifically, we design a conditional computation block that routes subproblems specialized neural architectures. Moreover, we propose preference-driee optimization algorithm that learns pairwise preferences between winning and losing solutions. We eval- uate the efficacy and versatility of POCCO by applying it to two state-of-the-at neural methods for MOCOPs. Experimental results across four classic MOCOP benchmarks demonstrate its significant superiority and strong generalization.
Introduction
Multi-objective combinatorial optimization problems (MOCOPs) involve optimizing multiple con- flicting objectives within a discrete decision space. They have attracted considerable attention from the computer science and operations research communities due their widespread applications in manufacturing [1]. logistics [25], and scheduling [17]. In such scenarios, decision makers must simul- tancously consider and balance multiple criteria, such cost, makespan, and environmental impact. Unlike single-objective combinatorial optimization problems (SOCOPs), which seek a single optimal solution, MOCOPs aim identify Pareto optimal solutions that reflect trade-off' among confficting objectives, making them inherently more challenging. Given their NP-hard nature, exact methods typically struggle solve MOCOPs within reasonable time frames, as computational complexity may increase exponentially with problem scale [13, 16]. Consequently, heuristic approaches have emerged as the main avenue for tackling MOCOPs. However, these heuristics often involve extensive iterative
Conresppnding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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local searches for each new instance, resulting in high computational costs. Furthermore, conventional heuristics often rely on extensive Nomain-specifee expertise and meticulous, problem-specific tuning. thus limiting their adaptability to broader classes of MOCOPs.
Recently, neural methods have achieved great success in solving SOCOPs [3, 8, 22, 23, 43, 60, 69, 70J by learning effective patterns of decision policies in data-driven way. Motivated by these advances, researchers have extended neural approaches MOCOPs, leveraging their advantages in bypassing labor-intensive heuristic design, accelerating problem solving through GPU parallelization, and flexibly adapting to diverse MOCOP variants. Typically, neural methods address MOCOPs by decomposing them into a set of scalarized subproblems, each a SOCOP defined by specific weight vector, and solving them using deep reinforcement learning (DRL) to approximate the Pareto front. Early approaches train or fine-tune a separate model for each subproblem using transfer learning or meta-learning techniques [34, 67]. However, these approaches require extensive computational resources and struggle to generalize to subproblems with unseen weight vectors. As an alternative, hypernetwork to modulate model parameters, allowing single model to address all subproblems. Nevertheless, it remains limited in effectively handling subproblems with diverse weight vectors. More recently, methods such as CNH [14] and WE-CA [6] have tackled this challenge by encoding weight vectors directly into the problem representations, resulting in a unified model that generalizes across various problem sizes. These methods are currently considered state-of- the-art (SOTA) in solving MOCOPs.
PMOCO [38] employs a weight-conditione
Current SOTA methods typically rely on a single neural network with limited capacity to handle all subproblems, which overcomplicates the learning task and results in suboptimal performance. straightforward solution to ease training and promote effective representation learning across subproblems is to increase the model capacity. However, determining how much additional capacity to allocate and where to introduce it within the architecture remains an open challenge. On the other hand, neural methods often adopt REINFORCE [56] as the training algorithm, relying solely on scalarized objective values as reward signals to guide policy updates. Given its on-policy nature, REINFORCE suffers from high gradient variance and lacks structured mechanisms for effective exploration [32]. These issues are exacerbated in MOCOP settings, where the vast combinatorial action space makes efficient exploration particularly difficult, ultimately hindering policy performance.
To address these issues, we propose POCCO (Preference-driven multi-objective combinatorial Optimization with Conditional COmputation), a plug-and-play framework that augments neural MOCOP methods with two complementary mechanisms. First, POCCO introduces a conditional computation block into the decoder, where sparse gating network dynamically routes each sub- problem through either a selected subset of feed-forward (FF) experts or a parameter- identity (ID) expert. This design enables subproblems to adaptively select computation routes (i.e., model structures) based on their context, efficiently scaling model capacity and facilitating more effective representation learning. Second, POCCO replaces scalarized rewards with pairwise preference learning. For each subproblem, the policy samples two trajectories, identifies the better one as the winner, and maximizes Bradley-Terry (BT) likelihood based the difference in their average log- likelihoods. Such comparative feedback guides the search toward policies that generate increasingly preferred solutions, enabling exploration of the most promising regions of the search space and more efficient convergence to higher-quality solutions. Our contributions are summarized as follows:
Conceptually, we address two fundamental limitations of existing approaches for solving MOCOPs: limited exploration within the vast solution space and the reliance single, capacity-limited model, which can lead to inefficient learning and suboptimal performance.
Technically, we propose conditional computation block that dynamically routes subprob- lems to tailored neural architectures. Additionally, we develop a preference-die algorithm leveraging implicit rewards derived from pairwise preference signals between winning and losing solutions, modeled using the BT framework.
Experimentally, we demonstrate the effectiveness and versatility of POCCO on classical MOCOP benchmarks using two SOTA neural methods. Extensive results show that POCCO not only outperforms all baseline methods but also exhibits superior generalization across diverse problem sizes.
2

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2 Related Works
Traditional methods for MOCOP. MOCOPs are significantly harder to solve than their single-objective counterparts. Classic cases such as the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) are already NP-hard, and adding multiple objectives only magnifies the difficulty [16]. Exact algorithms quickly become impractical for large instances or for problems that possess a vast Pareto set [18, 59]. Consequently, research has shifted toward heuris- tie methods that deliver high-quality Pareto fronts within reasonable time limits [25, 64]. Among these, multi-objective evolutionary algorithms (MOEAs) are widely used. They fall into two main categories: dominance- based MOEAs [11. 12, 46] and decomposition- based MOEAs [15, 26, 65]. Despite their popularity, MOEAs typically demand extensive manual design: practitioners must select and tune crossover, mutation, and selection operators along with many hyperparameters [50, 61, 63], and this abbor-intensive hand-engineering often limits overall performance.
Neural methods for MOCOP. Inspired by the success of DRL in SOCOPs [4], recent research extends neural methods to MOCOPs by solving a series of scalarized SOCOPs. These neural solvers generally follow two paradigms: one-to-one and one-to-many. The one-to-one paradigm trains or fine-tunes a separate neural model for each scalarized subproblem. Some approaches apply DRL algorithms individually and transfer parameters across models accelerate convergence [34, 57]. Others adopt architectures such as Pointer Networks [52] and Attention Models [29, 30), and optimize them using evolutionary strategies [47, 66]. To promote generalization across subproblems, meta- learning techniques are introduced [67]. However, this paradigm suffers from high training overhead and the burden of maintaining multiple models. In contrast, the one-to-many paradigm employs single neural network to handle all subproblems. This includes hypernetwork- based DRL frameworks that condition on weight vectors [35, 38, 58], and conditional neural heuristics that incorporate instance features, weight information, and problem size [6. 14].
Preference optimization. Preference optimization methods, such Direct Preference Optimiza- tion (DPO) [44] and Identity Preference Optimization (IPO) [2), have gained significant traction, particularly in large language model (LLM) training, due to their ability to directly optimize human preferences without relying on explicit reward modeling as required in reinforcement learning from human feedback (RLHF). These methods are typically designed for pairwise preference data, where human annotators identify preferred output over less preferred one in response to a given prompt. Another line of research explores simpler preference optimization objectives that do not depend on a reference model [21, 62]. Among them, SimPO [41] proposes using the average log- proobabbiil of generated sequence as an implicit reward. This approach aligns more closely with the model's generation process and improves computational and memory efficiency by eliminating the need for a separate reference model. Some studies [37] have explored applying preference optimization to SOCOPs, but its application to MOCOPs has been rarely investigated.
3 Preliminary
3.1 MOCOP
An MOCOP instance specified by data G can be formally defined as minTEx F(T)) (f)(), f2())...... ff (T)), where is the objective vector with k objective functions, π a feasible solution, and X' denotes the feasible solution space. Due to inherent conflicts among objectives, single solution that simultaneously optimizes all objectives typically does not exist. Instead, Pareto optimal solutions are sought represent different trade-offs, often guided by weight vectors, among competing objectives. We define the Pareto properties of the solutions as follows.
Definition 1 (Pareto Dominance). A solution π E X' is said to dominate another solution n' E X (i.e., क - x') if and only if f((T) S fi(='), Vi E {L, ,Ki,, and F(TT) & F(T').
Definition 2 (Pareto Optimality). A solution x²" E X is Pareto optimal if it is not dominated by any other solution. Accordingly. the Pareto set P is defined as all Pareto optimal solutions, i.e., p P= { E X $ ग E X < }.}. The Pareto front F is defined as images of Pareto optimal solutions in the objective space, i.e., F = (F(x") "(m²" E P).
Decomposition- based Methods. Since solving a SOCOP optimally is NP-hard, MOCOPs are signif- icantly more intractable due to the need to identify Pareto optimal solutions, the quantity of which
3

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grows exponentially with the problem size. Therefore, MOEAs are commonly adopted to compute approximate Pareto solutions. Among them, decomposition based MOEAs (MOEA/Ds) solve set of subproblems (i.e., SOCOPs) derived from the original MOCOP, a foundation underlying recent DRI-bbased neural methods. Specifically, the vanilla MOEA/D utilizes decomposition techniques to scalarize an MOCOP into N subproblems with set of uniformly distributed weight vectors
A2. ,NN). each of which satisfies X; (A ... , xF) A X Z 0 and ビ==
3.2 Subproblem Solving
Given subproblem (G, Ac), we formulate the solving process as Markov Decision Process (MDP). An agent iteratively takes the current state as input (e.g., the instance information and the partially constructed solution), and outputs a probability distribution over the nodes to be selected next. An action corresponds to selecting a node, either greedily or by sampling from the predicted probabilities. The transition involves appending the selected node to the partial solution. We parameterize the policy p by deep neural network 0, such that the probability of constructing complete solution is defined as po(T [G A;) IT== Po (T( T<(+ G, Az), where T is the total number of steps, and m and T<z represent the selected node and partial solution at the t-th step, respectively. The reward is defined as the negation of the scalarized objective, e.g., R(T) = Σه= ( with weighted sum decomposition. The policy network is typically optimized using the REINFORCE [56], which maximizes the expected reward L(olg, X²) (=) ((T)) via the following gradient estimator:
VaL(O Ac) = EE - bG))
(=
where the baseline function 6(-) reduces the gradient variance and stabilizes the training. There are two primary paradigms to extend the above approach solve MOCOPs. In the one-to-one paradigm, methods sequentially train separate models to solve subproblems with a predefined set of weight vectors. However, they often suffer from low training efficiency and generalize poorly to subproblems with unseen weight vectors. In contrast, the one-to-many paradigm trains single model to solve subproblems for arbitrary weight vectors. This typically achieved by incorporating a subnetwork that transforms each weight vector into model parameters, thus inducing tailored policies for each subproblem. Our POCCO follows the one-to-many paradigm and introduces two key innovations. Specifically, design conditional computation block to dynamically route subproblems to different neural architectures, and propose a preference-driven algorithm to train the model effectively.
4 Methodology
4.1 Overview
POCCO is a learning-based framework that trains a portfolio of policies to solve a set of scalarized subproblems {1)) obtained by decomposing MOCOP instance. Instead of forcing single policy to handle all subproblems, which often yields bland and suboptimal behavior, POCCO promotes specialization: each policy encouraged to focus a subset of subproblems, yielding a diverse policy ensemble. Such diversity is known to enhance multi-task optimization [53] by ex- panding the exploration space and ultimately improving solution quality. Technically, we achieve this diversity by activating different subsets of model parameters through a CCO block, enabling distinct computational paths emerge for different subproblems. Moreover, POCCO should encourage each policy to thoroughly explore the combinatorial solution space during training for reducing suboptimality. To achieve this, we replace raw rewards with preference signals. For each subprob- Training then
lem (G. A²), we construct set of Boaning-losingg solution pairs { xt ))) -- proceeds by maximizing the likelihood of the winning solutions while minimizing that of the losing ones, following a BT--style objective. This preference-driven training encourages the learned policy to explore the most promising regions of the search space, leading to more efficient convergence toward higher-quality solutions. Notably, POCCO is a generic, plug-and-play framework that can seamlessly involve different neural solvers for MOCOPs. We demonstrate this by augmenting two SOTA methods, CNH [14] and WE-CA [6], in Section 5.

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Probability
Probability
CCO Block
Compatibility Softmax
Compatibility Softmax
Normalization
Conditional Computation Block
FF
FF
FF
ID
h2
Multi-head Attention
Multi-head Attention
K
Q
K
V
Q
Router
P
(a) Backbone Decoder
POCCO Decoder (Ours)
Figure 1: Decoder structures of backbone and POCCO. Given an MOCOP instance with n + 1 nodes (c.g., ... customers and a depot, if applicable) and weight vector A; POCCO encodes their raw features into joint node embeddings {h₂ Fi=o using an encoder. At each decoding step t. the decoder forms query Q from the embeddings of the first and last selected nodes (T me), and computes the key K and value V via linear projections of (h,);=o The MHA layer processes Q, K, and V to produce context vector he, which is refined by the CCO block. The refined context is passed through a compatibility layer followed by a Softmax to compute the node selection probabilities. More details about the forward pass can be found in Appendix C.
4.2 Conditional Computational Block
Most approaches employ Transformer based architecture, where the encoder generates joint node embeddings { {h, V that capture the interaction between the instance G and the weight vector A, and the decoder produces candidate solutions conditioned on these embeddings. We propose CCO block to increase the model capacity and promote policy diversity across subproblems. To maintain efficiency, integrate the CCO block solely into the decoder of the backbone model. This design enables the generation multiple diverse solutions through a single, computationally expensive encoder pass, offering favorable trade-off between empirical performance and computational cost.
As illustrated in Fig. 1, the CCO block comprises multiple FF experts and single ID expert. We insert this block between the multi-head attention (MHA) layer and the compatibility layer in the decoder. Given a batch of MHA outputs { the CCO block dynamically routes each context vector h from the corresponding subproblem through either the FF or ID experts, forming distinct computation paths that function different policies. The ID expert allows the model to bypass the FF computation, promoting architectural sparsity and specialization [19]. Consequently, the CCO block facilitates the learning of dedicated, weight-specific policies tailored individual subproblems.
Formally, CCO block consists of: m FF experts {E1, Eg...., Em } with independent trainable parameters; 2) parameter-free identity expert Em++ 3) a router, implemented as gating network G parameterized by Wa. which determines how the inputs { are routed to the experts; and 4) a skip connection followed by an instance normalization (IN) layer. Given a single context vector h% let G(N²) E m= denote the output of the gating network, which represents the expert selection probabilities, and let E,(h) (h²) denote the output of the j-th expert. The output of the CCO block is:
m++1
G(L)),, + CCO(k = IN Σ
(2)
The sparse vector G(hh) activates only small subset of experts, either parameterized FF experts the parameter-free ID expert, thereby enabling diverse computation paths while reducing computational overhead. A typical implementation uses Top & operator that retains the & largest logits and masks the rest with co. In this case, the gating network output is: G(h)) = Softmax (Top k(h) Wa)).
Our proposed CCO block aligns with the principles of recent advances in efficiently scaling Transformer-basee models along both width [48] and depth [45]. specifie, combines mixture-of-expeers (MoE) layer, implemented using multiple FF experts to widen the network, with mixture-of-depths (MoD) layer, realized through an ID expert that allows inputs to skip compu- tation. Within the CCO block, each subproblem is adaptively routed only small subset of experts,
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Data Generation
Woight Ponao Port
Preference Driven MOCO
Instamoes
Weights
Construct MOCOP Subproblems
Obty
Explicit Reward (previous)
Implicit Reward (ours)
Obj
Reward
Solution
Solution
π W
Solution π
RL Policy
+2
BT Model
PL Policy
Obj 72
Sample
Figure 2: An overview of preference-driven MOCO, Unlike prior DRL methods that explicitly learn from scalarized rewards, our approach converts relative preferences into BT likelihood, providing an implicit reward signal to optimize the PL policy.
granting the model the expressiveness of significantly wider network while preserving computa- tional efficiency. As demonstrated in Section 5, this joint design achieves better capacity-effiiieny trade-off than scaling either dimension in isolation.
4.3 Preference-driver MOCO
To mitigate the exploration inefficiencies inherent in REINFORCE algorithms, we optimize relative preferences [41] instead of absolute objective values. We summarize our preference-driven MOCO in Appendix D and outline the complete pipeline in Fig. 2, which proceeds three steps as follows.
Generating preference pairs. For each scalarized subproblem (9, A,) in the training batch, the policy pe samples two candidate solutions, nh and x'. We denote nh" n' if nh is preferred over x', determined by the ordering of their scalarized objective values. This evaluator ranks the two solutions and designates the better one as the winning solution x and the other as the losing solution r'. A binary preference label y is then assigned, where y 1 if œ², and y 0 otherwise. This label serves as the supervision signal required for the preference-drived MOCO.
Defining an implicit reward. Distinct from DRL. training paradigms that rely raw objective values, POCCO treats the average log-likelihood of solution as an implicit reward fa, directly relating preferences between solutions their policy probabilities. This reward is inherently normalized by sequence length |=], thereby mitigating length bias between winning and losing solution pairs.
1 π
A.) = log (o(=
log po (,
Learning from pairwise comparisons. We formulate preference learning (PL) probabilistic binary classification problem through the BT model. Specifically, the BT model is a pairwise prefer- ence framework that uses function (-(-) (-) map reward differences into preference probabilities. assigns each solution strength proportional to its implicit reward (defined in Eq. (3)) and predicts the probability go that the winning solution outranks the losing one:
ge( < " A.) (33 [fo(=" (,) fa( (.)))))
(4)
where or( (-.) is the sigmoid function, and is fixed temperature that controls the sharpness with which the model distinguishes between unequal rewards. We maximize the likelihood the collected preferences, yielding the following loss function:
log po( G, A1) log po ( (g, Ai)
C(0)po. . G, Aj ", x²)x²) = y log (3
(5)
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In practice we collect multiple (x) pairs per update, sum their losses, and backpropagate through Pg. By maximizing the log-likelihood of go ud < o' g Az), the model is encouraged to assign higher probabilities to preferred solutions x" over less preferred ones x².
5
Experiments
5.1 Experimental settings
Training. We conduct extensive experiments to evaluate the effectiveness of the proposed POCCO across various MOCOPs, including multi-objective traveling salesman problem (MOTSP) [39], multi-objectivee capacitated vehicle routing problem (MOCVRP) [64], and multi-objective knapsack problem (MOKP) [23]] In MOTSP, the goal is to find a tour that visits all nodes exactly once, while minimizing multiple total path lengths, each computed based associated with a specific objective. Regarding MOCVRP, a fleet of vehicles with limited capacity must serve all customer nodes and return the depot, with the objectives of minimizing the total travel distance and the length of the longest individual route. As for MOKP, the problem involves selecting subset of items, each with weight and multiple objective-specifie values. The objective is to maximize all objective values simultaneously, while ensuring the total weight stays within the knapsack capacity. In this work, we consider three commonly used problem sizes: 20/50/1100 100 for MOTSP and MOCVRP, and n = 10002000 for MOKP.
distinct set of coordinates
Hyperparameters. We implement POCCO on top of two SOTA neural MOCO methods, CNH [14] and WE-CA [6], resulting in POCCO-C and POCCO-W, CO respectively. Most hyperparameters are aligned with those used in the original CNH and WE-CA implementations. Both models are trained for 200 epochs, with each epoch processing 100,000 randomly sampled instances and a batch size of B = 64. We use the Adam optimizer [28] with learning rate of 3 x 10-1 and weight decay of 10 10-6. The N weight vectors used for decomposition are generated following [10], with = 101 for and N 105 for K 3.
Baselines. We compare POCCO with a broad range of baseline methods across three categories, all employing weighted-sum (WS) scalarization ensure fair comparison: (1) Single-model neural MOCO approaches: This includes PMOCO [38], and recent SOTA methods CNH [14], and WE-CA [6]. Both CNH and WE-CA are unified models trained across problem size 100} (except n E (50, 51,-. ..... 200} for Bi-KP) (2) Multi-model neural MOCO
n E {20, 21, ................................ approaches: This category covers methods like DRL-MOA [34], MDRL [67], and EMNH [7]. Specifically, DRL-MOA trains a separate POMO model for each of the N subproblems, with the first model trained for 200 epochs and the rest fine-tuned for 5 epochs each using parameter transfer. MDRL and EMNH both initialize from a shared pretrained meta-model and fine-tune N subproblem- specific models using the same network structure and training settings as in [7]. (3) Non-learnable approaches, including classical MOEAS and other problem-specific heuristics: MOEA/D [65] and NSGA-II [12], each run for 4,000 iterations, serve as representative decomposition based and dominance based MOEAs, respectively. MOCOP-specine MOEAs such MOGLS [24], config- ured with 4,000 iterations and 100 local search steps per iteration, and PPLS/D-C [49], run for 200 iterations, are also considered. These methods use 2-opt heuristics for MOTSP and MOCVRP, and a greedy transformation heuristic [23] for MOKP. Finally, WS-LKH and WS-DP combine weighted-sum scalarization with powerful solvers, with LKH [20, 51] used for MOTSP and dynamic programming applied to MOKP.
Inference. We evaluate all methods using three metrics: average hypervolume (HV) [54], average gap, and total runtime per instance set. HV is widely used indicator in multi-objective optimization that reflects both the convergence and diversity of the solution set. A higher HV indicates better performance. To ensure consistency, HV values are normalized to the range [0. 1] using the same reference point for all methods. The gap is defined as the relative difference between a method's HV and the HV of POCCO-W. Methods with the "-Aug" suffix apply instance augmentation [38] to further improve performance. To evaluate statistical significance, we use the Wilcoxon rank-sum test [55] at a 1% significance level. The best results and others that are not significantly worse are marked in bold, while the second-best and statistically similar results are underlined. All experiments are implemented in Python and conducted on a machine with NVIDIA Ampere A A100-80GB GPUs and an AMD EPYC 7742 CPU. The code and dataset are publicly released for reproducibility.
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Table 1: Performance on BiTSP and MOCVRP Instances
BS-T0
Bi- Bi-TSP50
Bi- TSPI00
Method
HV
Gap
Time
HIV
Gap
Time
HV
Gap
Time
WS-LKH
0.6270
0.00%
10m
0.6415
0.05%
1.8h
0.7090
-0.17%
6h
MOEA/D
0.6241
0.46%
1.7h
0.6316
1.59%
1.8h
0.6899
2.53%
2.2h
NSGA-II
0.6258
0. 1.19
6.0h
0.6120
4.64%
6.1h
0.6692
5.45%
6.9h
MOGLS
0,6279
-0.14%
1.6h
0.6330
1.37%
3.7h
0.6854
3.16%
11h
PPLS/D-C
0.6256
0.22%
26m
0.6282
2.12%
2.8h
0.6844
3.31%
DRL-MOA
0.6257
0.21%
6s
0.6360
0.90%
9s
0.6970
1.53%
16s
MDRI.
0.6271
-0.02%
58
0.6364
0.84%
88
0.6969
1.54%
148
EMNH
0.6271
-0.02%
0.6364
0.84%
0.6969
1.54%
15s
PMOCO
0.6259
0.18%
68
0.6351
1.04%
128
0.6957
1.71%
26s
CNH
0.6270
0.00%
13s
0.6387
0.48%
168
0.7019
0.83%
33s
POCCO-C
0.6275
-0.08%
0.6409
0.14%
20s
0.7047
0.44%
42s
WE-CA
0.6270
0.00%
6s
0.6392
0.41% %
9s
0.7034
0.62%
18s
POCCO-W
0.6275
-0.08%
7s
0.6411
0.11%
14s
0.7055
0.32%
36s
MDRL-Aug
0.6271
-0.02%
47s
0.6408
0.16%
1.8m
0.7022
0.79%
5.4m
EMNH-Aug
0.6271
-0.02%
46s
0.6408
0.16%
1.8m
0.7023
0.78%
5.4m
PMOCO-Aug
0.6270
0.00%
39s
0.6395
0,36%
1.7m
0.7016
0.88%
5.8m
CNH-Aug
0.6271
-0.02%
1.3m
0.6410
0.12%
3.9m
0.7054
0.34%
12m
POCCO-C-AuR
0.6270
0.00%
2.2m
0.6416
0.03%
4.0m
0.7071
0.10%
14m
WE-CA-Aug POCCO-W-AR A-uu
0.6271
-0.02%
1.3m
0.6413
0.08%
3.6m
0.7066
0.17%
12m
0.6270
0.00%
2.2m
0.6418
0.00%
4.0m
0.7078
0.00%
14m
MOCVRP20
MOCVRP50
MOCVRP100
Method
HV
Gap
Time
HV
Gap
Time
HV
Gap
Time
MOFA/D
0.4255
1.07%
2.3h
0,4000
2.63%
2.9h
0.3953
3.33%
5.0h
NSGA-II
0.4275
0.60%
6.4h
0.3896
5.16%
8.8h
0.3620
11.47%
9.4h
MOGLS
0.4278
0.53%
9.0h
0.3984
3.02%
20h
0.3875
5.23%
72h
PPLS/D-C
0.4287
0.33%
1.6h
0.4007
2.46%
9.7h
0.3946
3.50%
38h
DRL-MOA
0.4287
0.33%
8s
0,4076
0.78%
12s
0.4055
0.83%
21s
MDRL
0,4291
0.23%
Ós
0,4082
0.63%
13s
0.4056
0.81%
22s
EMNH
0.4299
0.05%
7s
0,4098
0.24%
12s
0.4072
0.42%
22s
PMOCO
0.4267
0.79%
68
0.4036
1.75%
12s
0.3913
4.30%
22s
CNH
0,4287
0.33%
11s
0,4087
0.51%
15s
0,4065
0.59%
25s
POCCO-C
0.4294
0.16%
168
0.4101
0.17%
25s
0.4079
0.24%
53s
WE-CA
0.4290
0.26%
7s
0.4089
0.46%
10s
0.4068
0.51%
21s
POCCO-W
0.4294
0.16%
8s
0.4102
0.15%
17s
0.4084
0.12%
46s
MDRL-Aug
0.4294
0.16%
128
0.4092
0.39%
36s
0.4072
0.42%
2.8m
EMNH-Aug
0,4302
-0.02%
128
0.4106
0.05%
35s
0.4079
0.24%
2.8m
PMOCO-Aug
0,4294
0.16%
14s
0,4080
0.68%
42s
0.3969
2.93%
2.0m
CNH-Aug
0.4299
0.05%
21s
0.4101
0.17%
45s
0.4077
0.29%
1.9m
POCCO-C-Aug
0,4302
-0.02%
31s
0.4108
0.00%
1.4m
0.4086
0.07%
2.4m
WE-CA-Aug
0.4300
0.02%
15s
0.4103
0.12%
368
0.4081
0.20%
1.8m
POCCO-W-Aug
0.4301
0.00%
24s
0.4108
0.00%
1.2m
0.4089
0.00%
2.3m
5.2 Experimental results
Comparison analysis. The comparison results are presented in Table and Table 2. POCCO-W consistently achieves superior performance over WE-CA across all benchmark scenarios, establishing itself as the new SOTA results among neural MOCOP solvers. Similarly, POCCO-C outperforms CNH in every case. Both variants also surpass their augmentation-basee and CNH-Aug, on Bi-TSP20 and Bi-CVRP100, highlighting POCCO's enhanced ability to explore the solution space and approximate high-quality Pareto fronts. When further combined with instance augmentation, POCCO demonstrates additional performance gains. Please note that POCCO with instance augmentation yields lower HV values compared with its non-augmented counterpart on Bi- TSP20. This is because decomposition- based methods focus on optimizing individual subproblems rather than ensuring overall solution diversity. While augmentation improves solution quality for specific subproblems, it may reduce the number of non-dominated solutions, resulting in smaller HV. Compared with multi-model approaches that require training or fine-tuning separate models for each subproblem, POCCO delivers superior results while maintaining single shared model. Notably, POCCO achieves better results on Bi-TSP50 than WS-LKH, setting where previous neural solvers have consistently failed. In terms of efficiency, POCCO significantly reduces computational time.
counterparts, WE-CA-Aug
8

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Table Performance comparison on Bi-KP and Tri-TSP Instances
Method
Bi-KP50
Bi-KPI00
Bi-KP200
HV
Gap
Time
HV
Gap
Time
HV
Gap
Time
WS-DP
0.3561
0.03%
22m
0,4532
0.04%
2h
0.3601
0.06%
5.8h
MOEA/D
0.3540
0.62%
1.6h
0.4508
0.57%
1.7h
0.3581
0.61%
1.8h
NSGA-II
0.3547
0.42%
7.8h
0,4520
0.31%
8.0h
0.3590
0.36%
8.4h
MOGLS
0.3540
0.62%
5.8h
0.4510
0.53%
10h
0.3582
0.58%
18h
PPLS/D-C
0.3528
0.95%
18m
0.4480
1.19%
47m
0.3541
1.72%
1.5h
DRL-MOA
0.3559
0.08%
8s
0.4531
0.07%
15s
0.3601
0.06%
32s
MDRL
0.3530
0.90%
7s
0.4532
0.04%
18s
0.3601
0.06%
358
EMNH
0.3561
0.03%
7s
0.4535
-0.02%
178
0.3603
0.00%
48s
PMOCO
0.3552
0.28%
8s
0,4523
0.24%
22s
0.3595
0.22%
50s
CNH
0.3556
0. 17%
168
0.4527
0.15%
23s
0.3598
0.14%
558
POCCO-C
0.3560
0.06%
20s
0.4535
-0.02%
368
0.3603
0.00%
1.4m
WE-CA
0.3558
0.11%
8s
0,4531
0.07%
16s
0.3602
0.03%
50s
POCCO-W
0.3562
0.00%
lls
0.4534
0.00%
268
0.3603
0.00%
1.3m
Tri TSP20
Tri-TSP50
Tri-
Method
HV
Gap
Time
HV
Gap
Time
HV
Gap
Time
WS-IKKH
0.4712
0.00%
12m
0,4440
-0.0%%
1.9h
0.5076
-0.55%
6.6h
MOEA/D
0.4702
0.21%
1.9h
0.4314
2.77%
2.2h
0.4511
10.64%
2.4h
NSGA-II
0,4238
10.06%
7.1h
0.2858
35.59%
7.5h
0.2824
44.06%
9.0h
MOGLS
0.4701
0.23%
1.5h
0,4211
5.09%
4.1h
0.4254
15.73%
13h
PPLS/D-C
0.4698
0.30%
1.4h
0,4174
5.93%
3.9h
0.4376
13.31%
14h
DRL-MOA
0.4699
0.28%
6s
0.4303
3.02%
9s
0.4806
4.79%
18s
MDRL
0.4699
0.28%
5s
0.4317
7.70%
10s
0.4852
3.88%
178
EMNH
0.4699
0.28%
0.4324
2.55%
0,4866
3.61%
17s
PMOCO
0.4693
0.40%
5s
0,4315
2.75%
12s
0.4858
3.76%
338
CNH
0,4698
0.30%
10s
0,4358
1.78%
14s
0.4931
2.32%
25s
POCCO-C
0,4704
0.17%
18s
0,4393
0.99%
17s
0.4985
1.25%
28s
WE-CA
0.4707
0.11%
5s
0.4389
1.08%
8s
0.4975
1.45%
17s
POCCO-W
0.4710
0.04%
0.4397
0.90%
0.4985
1.25%
238
MDRL-Aug
0.4712
0.00%
4.2m
0,4408
0.65%
25m
0.4958
1.78%
1.6h
EMNH-AUS
0.4712
0.00%
4.2m
0,4418
0.43%
25m
0,4973
1.49%
1.6h
PMOCO-Aug
0.4712
0.00%
4.9m
0.4409
0.63%
28m
0.4956
1.82%
1.6h
CNH-Aug
0.4704
0.17%
8.5m
0,4409
0.63%
28m
0.4996
1.03%
1.6h
POCCO-C-Aug
0.4706
0.13%
8.9m
0,4419
0.41%
34m
0.5023
0.50%
2h
WE-CA-Aug 0.4712
0.00%
8.2m
0.4432
0.11% 1%
29m
0.5035
0.26%
1,7h
POCCO-W-Aug
0.4712
0.00%
8.9m
0.4437
0.00%
33m
0.5048
0.00%
2h
For example, POCCO-W-Aug solves Bi-TSP100 in only 14 minutes, while WE-LKH requires about 6.0 hours, with POCCO delivering comparable solution quality.
Rut-of-distributio size generalization analysis. We evaluate the generalization ability of the models on out-of-distribution, benchmark instances KroAB 100/150/200 [40]. All neural methods trained Bi-TSP100. except for CNH, WE-CA, and POCCO, which are trained across varying sizes n E (20, 21, 21, 100). The results, visualized in Fig. 3, show that POCCO- SOCCO-W-Aug W-Aug consistently achieves the best generalization performance compared with other neural baselines. POCCO-C C-Agg also outperforms CNH-Aug across all evaluated scenarios. Full benchmark results and experiments on larger-scale instances Bi-TSP150020 are provided in Appendix I and Appendix J, respectively.
Effectiveness of PL. We assess the training efficiency of PL by comparing it to REINFORCE on the WE-CA and POCCO-W W models using the Bi-TSP100 dataset. As shown by the validation curves in Fig. 4a, PL achieves faster convergence despite identical network architectures. Notably, for WE-CA, training with PL for 100 epochs reaches performance comparable 200 epochs of REINFORCE. Similar improvements are observed for POCCO-W. These results demonstrate that PL accelerates the training process and achieves better performance with fewer training epochs.
Effectiveness of the CCO block. To evaluate the impact of the CCO block's structure and placement, we compare POCCO-W with WE-CA and several POCCO variants: POCCO-E (CCO inserted the encoder replacing the FF layer), POCCO-D (CCO replacing the final linear layer of MHA in the decoder, using MLP experts), POCCO-ME (replacing CCO with standard MoE layer), POCCO-MD (replacing CCO with three MoD layers), POCCO-MED (using MoF. in the encoder and MoD in place of CCO in the decoder), and POCCO-Emp (replacing the identity expert in CCO

---

CHE (100 CUNON
PODCSSCAAE Colee ITV (999)
24
18
(a) K1oAB100
(b) Kr0AB150
(c) KroAB200
Figure 3: Pareto fronts of benchmark instances.
0.305
0313
2318
0736
3351
2
0331
WE-CA-PL POCOO-N-RL
7333
POCCO-N
resionad
Apo
eorcon moa
Petcor
226
(a) Validation curves
(b) Bi- ISP00
(c)
TSP200000
Figure 4: Ablation study:(a) validates the effectiveness of PL: (b) and (c) verify the effects of different CCO block variants.
with empty expert). As shown in Fig. 4b, all variants outperform WE-CA the in-distribution Bi-TSP100, with POCCO-W W, POCCO-ME, and POCCO-Emp achieving the most notable gains. the out-of-distribution Bi-TSP200 in Fig. 4c, only POCCO-W W and POCCO-Emp maintain strong performance, while POCCO-ME performs worst, even underperforming WE-CA. These results highlight the importance of both the structure and placement of the CCO block for achieving strong generalization across in- and -iit---diitiittionn settings.
Hyperparameter study. We conduct experiments to examine the impact of different key hyperpa- rameters on POCCO's s performance. As detailed in Appendix L, the number of CCO block layers, the number of experts, the Top k value, and the temperature parameter 3 all influence model effectiveness. For the problems studied, the desirable settings, as identified based on empirical results, are: one CCO block layer, four FF experts and one ID) expert, Top k 2, B 3.5 for bi-objective tasks, and 4.5 for tri-objective tasks.
6 Conclusion
This paper presents POCCO, a plug-and-play framework tailored for MOCOPs, which adaptively routes subproblems through different model structures and leverages PL for more effective training. POCCO integrated into SOTA neural solvers, and extensive experiments demonstrate its effectiveness. Ablation studies further highlight the necessity of both CCO block and Pl., and reveal the critical impact of the design and placement of the CCO block. We acknowledge certain limitations, such as the limited capability to address real-world MOCOPs with complex constraints or large problem sizes. Addressing these challenges may require constraint-handling mechanisms [5] or divide-and-conquer [36] strategies, which we leave for future work.
Acknowledgments and Disclosure of Funding
We thank the anonymous reviewers and (S)ACs of NeurIPS 2025 for their constructive comments and dedicated service to the community.
10

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Appendix
A Details of MOCOP
Here we elaborate on the problem definitions for the three typical MOCOPs, i.e., MOTSP, MOCVRP, and MOKP.
MOTSP. An MOTSP instance involves multiple cost matrices, aiming to identify set of tours, i.c. node sequences, that are Pareto optimal. For instance, a k-objective TSP instance 9 with n + nodes is featured by cost matrices C* = (c) (,x)), with i E {I,-. ,.) and j. k E (0....,,) .n]. The objectives are defined as follows,
min F(T) = min( (TT)) (T(T)), *EX
n- 1
with f((x) -0 5o
+
where π -TO; .T. TT) with ny E {0, (0...., n). X' represents all feasible solutions (i.e., tours),
ensuring each node is visited exactly once. This paper considers the Euclidean MOTSP following [16, 34, 38]. Each node has a 2x-dim feature vector 0j [loc], locz. ... ,1oc%). where loc, R is the coordinate for the i-th objective. The objective ((T is defined as f((T) = |loct,. - -loct ll2 +
Ejzo |lock loct x lla.
MOCVRP. An MOCVRP instance consists of n customer nodes and one depot node. Each node has 3-dim feature vector oj |loc,, 6z]. where loej and ő,. for j E {o..... .n.. correspond the coordinates and demand of node j. Notably, for the depot node, Ou = [loca, do]. the demand óy is serve all customers in multiple
set to O. Vehicles with a capacity of (Q 5,) are employed routes, with each route commencing and concluding at the depot. The problem must satisfy the following constraints: 1) Each customer visited exactly once, and The total demand of customers in each route must not exceed the vehicle's capacity. In our study, we focus on the bi-objective CVRP, aligning with prior research [31, 38). Specifically, we aim to minimize two objectives: the total length of all routes and the length of the longest route (i.e., the makespan).
МОKP. An MOKP instance consists of + items, where each item j characterized by a 2-dim feature vector Oj- [Wj.Pj]. representing its weight wy and profit vector Pj. for E {0.... .n.. The profit vector Pj E I2 corresponds
distinct objective values associated with item j. knapsack with capacity of C is provided, where C > Wy for each item. The goal is to select subset of items to place into the knapsack while satisfying the following constraint: the total weight of selected items must not exceed the knapsack capacity C. In our study, we focus on the bi-objective knapsack problem, consistent with previous research [40, 68). Specifically, we aim to simultaneously maximize two objectives: the sum of the first and second profit components across the selected items.
B Theoretical Analysis
We derive the loss function and gradient for our PL objective and contrast it with the REINFORCE gradient. Our theoretical (and empirical) analyses demonstrate that PL yields significantly lower gradient variance, which contributes to faster and more stable convergence.
B.1 Formulation of Loss Function
Different from the RL, our PL method exploits the preference relations between generated solutions according their objective. The explicit preference f (T( A) is defined as the negation of the scalarized objective. We contruct preference pair, denoted as (π" x²), along with binary preference label y, where y 1 if n "0 & at (i.e., (" X) f (n (m(6 [G, X)), and y 0 otherwise.
22

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For the policy p A) used to construct solution its implicit preference is defined as the average log-likelihood: fT )) = [ log Pe ( 9, X). For a preference pair (," *",, the preference distribution is modeled using the Bradley-Tery (BT) ranking objective and implicit preferences:
go (
[1 = [ 99 [T( (9,,)l),
where a--) ) is the sigmoid function, and 8 > 0 is fixed temperature that controls the sharpness with which model distinguishes between unequal rewards. By maximizing the log-likelihood of go(n) (T < of G, A), the model is encouraged to assign higher probabilities to preferred solutions n compared with less preferred solutions yf. We can derive the PL. loss function:
(Pr(O0pp 9, "" y log o (3) log (
9,
log pe (TT G. A)
|T
PL directly distinguishes optimization signals based on exact preferences, and the strength of the optimization signal correlates with the difference in log-likel,oood, requiring the model to maximize the probability gap between π and r².
B.2
Gradient Analysis
Let denote the argument of the sigmoid function:
log ( X)
log po ( Gg, {) n
31
The gradient of LPl with respect is:
0LPL Voz.
VoLPL
8z
Derivative of -y log (2) becomes:
dCPL
y(11 -z))).
Oz
Gradient of with respect to 0:
V02 = 3] m
V log po 19,
Vo log log
Combining these, the total gradient becomes:
1 |x
T [2))]
T V lo
V log Po (g, A)].
The gradient increases the likelihood of T and decreases the likelihood of न.. For comparison, we analyze the loss function in the REINFORCE, algorithm here:
( A) = (R((x) - (a]logo(T[9,))))
where 6 is baseline to distinguish positive negative optimization signals for each solution и. The gradient of the REINFORCE algorithm is:
VoLas. (R(m)) -b) Vologno(=IS, X)-
REINFORCE relies on absolute returns R(T), whose large variance propagates directly to the gradient. In contrast, LPL uses relative returns inside sigmoid, yielding the difference of two normalized log-like likeeeiiooodds. This pairwise, centered signal reduces gradient variance and produces smoother, more stable updates, leading to faster and more reliable convergence. This theoretical insight is further supported by our empirical analyses. We provide concrete evidence this in Table which reports the average gradient variance in the first five batches of training POCCO-W under both RI. and PI. frameworks. Across three MOTSP settings, PI. consistently exhibits two to four orders of magnitude lower variance, leading to faster convergence and more stable optimization dynamics.
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Table 3: Gradient variance in REINFORCE vs. preference learning.
Problem Size
Batch
RL Variance
PL Variance
MOTSP20
0.054648
0.000314
2
0.039951
0.0002552
3
0.019038
0.000191
0.009093 4
0.000115 15
0.010742 5
0.000104
MOTSP50
0.474784
0.000142
2
0.220530
0.000077
3
0.140275
0.000048
4
0.092114
0.000040
0.065286 5
0.000031
MOTSPI00
1
10.124832
0.000059
6.355499
0.000039
3
3.461700
0.000032
1.755804
0.000021
5
1.111246
0.000014
C MOCOP Solvers with POCCO
Existing neural MOCOP solvers are based on a transformer-based architecture that consists of an encoder and decoder. The encoder is used to generate embeddings for all nodes based on the instance G and the weight vector The decoder is used to decode a sequence actions based on these embeddings in an iterative fashion. To demonstrate the versatility of our POCCO, we integrate into two SOTA neural methods, WE-CA [6] and CNH [14], yielding POCCO-W and POCCO-C.
C.1 POCCO-W
Given an instance G comprising n + nodes with Z-dimensional features {o. C. R and weight vector A, first obtains initial embeddings by applying separate linear projections the node features and the weight vector:
h W o +b, ViE .o)....n). =
where Wo E I2 dd W E Raxk, and b", bA Rd are learnable parameters, with embedding dimension d = 128. POCCO-W integrates the weight embedding into each node embedding in feature-wise fashion within the encoder. To ensure harmonious interaction, the weight and node embeddings are updated simultaneously. The encoder itself comprises layer applying conditional attention sublayer, followed by residual Add & Norm (skip connection with instance normalization), fully connected feed-forward sublayer, and a second Add Norm.
transformer layers, each
Specifically, the conditional attention sublayer first conditions node embeddings on the weight embedding via a feature-wist affine transform:
(8) where 7 and 3 are trainable matrices; denotes element-wise multiplication. Then, the weight and node embeddings updated via the Multi-Head Attention (MHA) mechanism with 8 heads and an Add & Norm, as follows:
W M/- B = W M M 3, Vi .o....,n).
he (N + MHA (A 16. M,)));
h, (N + MHA(H, (M/-1, (o....,K(2) Vie m..
(10)
Afterwards, fully connected feed-forward sublayer and another Add & Norm are employed to yield the weight embedding Nt and the node embeddings {h] as follows:
A IN + FF(h))),
(11)
1/ (h + FF()x)),
(12)
24

---

Given the eventual node embeddings {h, and weight embedding hy output by the encoder, the decoder autoregressively computes the probability of node selection over T steps. At decoding step t E (I...., 7), the advanced context vector he is produced by an MHA layer with 8 heads based on the context embedding V/ and eventual node embeddings as follows:
be = MHA (Ves {1x,, ho...., 1nn
(13)
Context embedding. For MOTSP, the context embedding Ve is obtained by concatenating the embeddings of the first and last visited nodes, and all previously visited nodes are masked when computing selection probabilities. together with the remaining vehicle capacity, while nodes that have already been visited or whose demand exceeds the remaining capacity are masked. In MOKP, the context embedding combines the graph embedding h = Ei=o hy with the remaining knapsack capacity, masking both items that are already selected and those whose weight exceeds the remaining capacity when computing selection probabilities.
MOCVRP, v, consists of the embedding of the last visited node
The context vector he is fed through the CCO block to produce a glimpse ge based on Eq. 2 (i.c., Ge ())( This glimpse 9e is then used in a compatibility layer compute unormalized compatibility scores or as follows:
{c. (1
-00, C. ta.na 8o (W*)a)) )
if node i is masked,
a
(14)
otherwise,
va
where C' is set to 50 and W is learnable weight matrix. Finally, the node-selection probability for the scalarized subproblem is given by:
Po(TTe #14-1,8) Softmax(a).
(15)
C.2 POCCO-C
POCCO-C mirrors the overall design of POCCO-W but (i) replaces every aaddtiionllaatentioon sublayer in the encoder with a dual-attention sublayer and (ii) enriches the context vector he via problem-size embedding (PSE) layer. Specifically, each of the L encoder layers in POCCO-C C-C consists of dual-attention sublayer, followed by residual Add & Norm, fully connected feed-forward sublayer, and a second Add & Norm. Concretely, at layer the dual-attention and first Add & Norm operate as:
hy IN + MHA(M 111 ,16 1/ ())); R. = MHA (M4-1 (1/-1 1 "..,..,,,,") + MHA (4/-1, (M) ViE (o........
(16)
(17)
h. -IN
Vie mo.
(18)
Besides, POCCO-C employs the sinusoidal encoding based on sine and cosine functions to yield the PSE as follows,
PSE(C, 21) sin(E/1000002/44
(19)
PSE(E. 2i + 1) cos(E/ cos(E/1000002//))
where <(E = n + 1) and i (i (O.... ,63}) mean the problem size and dimension. The resulting
d-dimensional PSEs are then processed using two linear layers with trainable matrices Wş1 E Rdx2d and Wga 2ddxd, POCCO-C injects the size information by adding the results from PSE to [ from the encoder, such that,
hy = h2 + hg. with dg = (PSE(E, Wee
(20)
Then, POCCO-C produces the advanced context vector he based on the context embedding ve and the size-injected node embeddings (h. ( through a MHA layer with 8 heads as follows,
he MHA Ver (16. .6,....,55))
(21)
25

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Algorithm Preference-driven MOCO
Input:
Instance distribution G. weight vector distribution A, number of training steps E, batch size B. number of tours per subproblem:
Output: The trained policy network 0:
1: Initialize policy network 0.
to E do 2: for SAMPLEEWIIGTVVECTRRCA 96 SAMPLEINSTANCE (9), vb E {1. B}
3: Àg
4: 7..0 -SAMPLESSSLUUTOOSSSS((( Ab)). Vj (1,... K}, v (1,... B}
5: VS,P ~ PAIRVISEPREEEEEENCCC (acA)) Vi.p (1,...K) vb E -1. B}
6: Calculate gradient Ve((() according to Eq. (5)
7: B ADAM(B, VoL(O))
8: end for
C.3 Gating Mechanism
Our CCO block employs subproblem-leve gating mechanism to dynamically route each context vector he subset of experts from pool consisting of FF experts and ID expert (thus 4). Let d be the hidden dimension and WG (m++) denote the trainable gating weight matrix. Given a batch of context vectors X = BB b=1 E IR Bxd where is the batch size, the gating network computes a score matrix: H = X We E R (x(+1), where each element Hy' represents the affinity or preference (score) of subproblem towards expert.j.
For each subproblem, the router selects the Top-k highest-scoring experts from the score vector Hig These selected experts are then activated, and their outputs are weighted by poftmax-normalizes version of their scores. In our POCCO with k 2, each subproblem is routed its two highest- scoring experts.
This Top-k routing plays critical role by reducing computation, as only l experts are activated per subproblem (i.e., sparse activation widely used in MoE structure [69]). also encourages expert specialization, since different subproblems (defined by context and weight vectors) tend to activate different subsets of experts. Additionally, allows the inclusion a parameter- free ID expert, which often selected when minimal transformation is needed. This enables the router learn when it is appropriate to leave representation unchanged.
D Training Algorithm
The training algorithm is provided in Algorithm 1. To train the model with preference learn- ing, first sample batch of instances and weight vectors {GG,, A (as in Line 3). Then, for each scalarized subproblem (G,, Ap), we construct a set winning-losing solution pairs (m4b, nPP.),, Vj.P E (1..... K} (as in Lines 4-5). Training then proceeds by maximizing likelihood of the winning solutions while minimizing that of the losing ones (as in Lines 6-7).
E Decomposition Approaches
The major decomposition techniques include weighted-sum, Tchebycheff, and penalty-based bound- ary intersection (PBI) approaches [42, 65], respectively.
Weighted-sum Approach. Given an MOCOP, the ith subproblem (i.e., SOCOP) is defined with the äth weight vector \, such that,
min (T/()) Σx- (.)()) TEX
(22)
Tehebycheff Approach. It minimizes the maximal distance between objectives and the ideal reference point, such that,
min ge(m|Ai,,") = max { - -2]1]. EEX
(23)
18iSK
26

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where 2" (2f. ....... 2x signifies the ideal reference point with 2j - min X). It guarantees that the optimal solution in Eq. (23) under specific (but unknown) weight vector A, could be a Pareto optimal solution [9].
PBI Approach. This approach formulates the ith subproblem of an MOCOP as follows,
min
gp( A) = dp + ad2
where di [+(
(24)
d2 - (x= T
where a >> is preset penalty item and x is the ideal reference point as defined in the Tchebycheff approach.
F Hypervolume
Hypervolume (HV) is a widely used indicator to evaluate approximate Pareto solutions MOCOPs. Formally, the HV for a set of solutions P is defined as the volume of the subspace, which is weakly dominated by the solutions in and bounded by reference point r", such that,
HV(P) = (fr E B E P, <r>r")),
(25)
where Şd denotes the Lebesgue measure on the k-diimnnsional space, i.e., the volume for a m- dimensional subspace [54]. Since the range of objective values varies among different problems, we report the normalized HV HP)) HV(P)/ I Ir] =:/. where the ideal point = (z1,..., =x) satisfies < min (f((T) π E P) (or
E P) for maximization). Vi E {1....,k] All methods share the same r" and for an MOCOP, as given in Table 4, and we report the average (P) over all test instances in this paper.
max(f((=))
.K..
Table 4: Reference points and ideal points.
Problem
Size
r*
(20, 20)
(0,0)
50
(35,35)
(0,0)
Bi-TSP
100
(65, 65)
(0,0)
150
855
(0,0)
200
(115, 115)
(0,0)
20
(30,4)
(0.0)
Bi-CVRP
50
(45,4) 4)
(0,0)
100
(80,4)
(0,0)
50
(5,5)
(30,30)
Bi-KP
100
(20, 20)
(50,50)
200
30)
(75,75)
20
(20, 20. 20)
(0.0)
Tri-TSP
50
(35, 35, 35)
(0.0)
100
(65, 65, 65)
(0,0) 0)
G Instance Augmentation
To further improve the performance of POCCO at the inference stage, we apply the instance augmentation proposed in [30], which also used in PMOCO [38]. The rationale of in- stance augmentation is that an instance of Euclidean VRPs can be transformed into different ones that share the same optimal solution, e.g., by flipping coordinates for all nodes in an in- stance. Given coordinate (I, v) in VRP there are eight simple transformations, i.e., (x', (e'3)) (x., y); (y. æ); (x, 1-y); (v, 1-1r;; -z, (): (1,,, (1 xr, 1 1- y); (1 y-y, 1 1-x). our paper, we adopt these transformations for each objective, respectively. Hence, we could have 8 transformations for Bi-CVRP (since there is only one coordinate for each node), 82 64 transformations for Bi-TSP. and g = 512 transformations for Tri-TSP, respectively.
27

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H Impact of Neutralizing Non-Dominated Pairs
Our method adopts a decomposition- framework, where the MOCOP divided into set of scalarized subproblems, each associated with weight vector. During training, the goal is to find high-quality solutions with respect to each subproblem, rather than globally exploring the entire Pareto front once. In this context, preference learning operates at the subproblem level. We acknowledge, however, that sampled solutions may occasionally mutually non-dominated in the original objective space. To investigate the effect of this scenario, conducted an additional experiment where preference signals for such non-dominated pairs were set to O, effectively treating them as equivalent. The results, shown in Table 5, indicate that this modification (NDS) significantly degrades model performance across all Bi-TSP instances.
This observation suggests that introducing indifference signals for non-dominated pairs hinders the learning process. One reason is that reduces the amount of effective preference supervision per subproblem, especially in early training stages when many solutions are of similar quality. In contrast, using scalarized values provides consistent and differentiable training signal aligned with decomposition objective. Therefore, although scalarized comparisons may not reflect global Pareto dominance in every case, they remain well-juu well-justified and necessary within the decomposition- based learning paradigm, and do not appear to bias the overall Pareto front approximation based empirical findings.
Table 5: Performance on Bi-TSP instances when neutralizing non-dominated pairs.
Problem Size
POCCO-W HV
NDS HV
Bi-TSP20
0.6275
0,5516
Bi-TSP50
0.6411
0,5160
Bi-TSPI00
0.7055
0.5601
Bi-TSP150
0.7033
0.5494
Bi-TSP200
0.7371
0.5809
I Detailed Results on Benchmark Instances
The detailed outto--distribution generalization results are presented in Table further confirming the exceptional generalization ability of our POCCO.
Experimental Results on the Larger Problem Sizes
The results on Bi-TS0,50/200, summarized in Table 7, show that POCCO-W consistently achieves the best generalization performance compared to all neural baselines and classical MOEAs.
Furthermore, to demonstrate scalability, we include experimental results on even larger problems, including Bi-TSP with 300/500/1000 nodes and MOKP with 300/500/1000 items. These results summarized Table and Table 9. As shown, POCCO-W consistently outperforms the baseline WE-CA in most large-scale cases.
K
Comparison with DPO
We have added comparative experiment between preference learning (PL) method and the recent DPO objective, as shown in Table 10. Since DPO requires two models, i.c., a policy model and reference model, we use the same architecture and initialization for both, with the reference model updated from the previous epoch. This setup significantly increases training time for DPO (122h 36h for POCCO-W).
As shown Table 10, while DPO achieves slightly better performance on the smallest instance (MOTSP20), it is consistently outperformed by POCCO-W on larger instances (MOTSP50-20) These results suggest that our PI. method offers more favorable trade-off in terms of both training efficiency and solution quality, especially for large-scale problems.
28

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Table 6: Performance on KroAB Instances
KroAB100
KroAB150
KroAB200
Method
HIV
Gap
Time
HV
Gap
Time
HV
Gap
Time
WS-LKH
0.7022
-0.23%
2.3m
0.7017
-0.59%
4.0m
0.7430
-0.83%
5.6m
MOEA/D
0,6836
2.43%
5.8m
0.6710
3.81%
7.1m
0.7106
3.57%
7.3m
NSGA-II
0.6676
4.71%
7.0m
0.6552
6.08%
7.9m
0.7011
4.86%
8.4m
MOGLS
0.6817
2.70%
52m
0.6671
4.37%
1.3h
0.7083
3.88%
1.6h
PPLS/D-C
3.15%
38m
0.6659
4.54%
0.7100
3.65%
3.8h
DRL-MOA
0.6903
1.47%
10s
0.6794
2.61%
12s
0.7185
2.50%
18s
MDRL
0.6881
1.78%
0.6831
2.08%
0.7209
2.17%
16s
EMNH
0.6900
1.51%
9%
0,6832
2.06%
11s
0.7217
2.06%
16s
PMOCO
0.6878
1.83%
0.6819
2.25%
0.7193
2.39%
17s
CNH
0.6947
0.84%
16s
0.6892
1.20%
19s
0.7250
1.1%%
22s
POCCO-C
0.6965
0.59%
30s
0.6925
0.73%
40s
0.7302
0.91%
50s
WE-CA
0.6948
0.83%
9s
0.6924
0.75%
12s
0.7317
0.71%
16s
POCCO-W
0.6981
0.36%
0.6946
0.43%
0.7345
0.33%
40s
MDRL-Aug
0.6950
0.80%
10s
0.6890
1.23%
16s
0.7261
1.47%
25s
EMNH-AUg
0.6958
0.69%
0,6892
1.20%
0.7270
1.34%
25s
PMOCO-Aug
0.6937
0.98%
11s
0.6886
1.29%
18s
0.7251
1.60%
30s
CNH-Aug
0,6980
0.37%
17s
0.6938
0.54%
26s
0.7303
0.90%
37s
POCCO-C-Aug
0.6999
0.10%
32s
0.6959
0.24%
48s
0.7334
0.47%
1.1m
WE-CA-Aug
0.6990
0.23%
10s
0.6957
0.27%
20s
0.7349
0.27%
31s
POCCO-W-Aug
0.7006
0.00%
22s
0.6976
0.00%
39s
0.7369
0.00%
59s
Table 7: Performance on Bi-TSP150 and Bi-TSP200 Instances
Bi-TSP150
Bi-TSP200
Method
Gap
Time
HV
Gap
Time
WS-LKH
0.7149
-1.33%
13h
0.7490
-1.23%
22h
MOEA/D
0.6809
3.58%
2.4h
0.7139
3.51%
2.7h
NSGA-11
0.6659
5.71%
6,8h
0.7045
4.78%
6.9h
MOGLS
0.6768
4.16%
22h
0.7114
3.85%
38h
PPLS/D-C
0.6784
3.94%
21h
0.7106
3.96%
32h
DRL-MOA
0.6901
2.28%
36s
0.7219
2.43%
1.2m
MDRL
0.6922
1.98%
368
0.7251
2.00%
1.1m
EMNH
0.6930
1.87%
37s
0.7260
1.88%
1.1m
PMOCO
0.6910
2.15%
42s
0.7231
2.27%
1.3m
CNH
0.6985
1.09%
50s
0.7292
1.45%
1.4m
POCCO-C
0.7011
0.72%
1.5m
0.7333
0.89%
2.5m
WE-CA
0.7008
0.76%
45s
0.7346
0.72%
1.3m
POCCO-W
0.7033
0.41%
1.4m
0.7371
0.38%
2.4m
MDRL-Aug
0.6976
1.22%
37m
0.7299
1.35%
1.1h
EMNH-Aug
0.6983
1.12%
39m
0.7307
1.24%
1.1h
PMOCO-Aug
0.6967
1.35%
40m
0.7283
1.57%
1.2h
CNH-Aug
0.7025
0.52%
41m
0.7343
0.76%
1.2h
POCCO-C-Aug
0.7043
0.27%
55m
0.7366
0.45%
1.5h
WE-CA-Aug
0.7044
0.25%
42m
0.7381
0.24%
1.2h
POCCO-W-Aug
0.7062
0.00%
45m
0.7399
0.00%
1.4h
L Hyperparameter Study
Effects of the J. 5 shows how the temperature parameter 3 in the preference learning loss affects performance (HV) across four benchmark tasks. A moderate value of I consistently yields the best results. For the three bi-objective problems (Bi-TSP100, MOCVRP100, Bi- Bi-KP))0), performance peaks around
3.5; larger or smaller values provide no additional benefit, and in Bi-KP100, an performance improves up to B = 4.5 and then declines slightly. Overall, these trends justify the default settings adopted in our experiments:
overly large 3 = 5 sharply degrades HV. For the tri-objective problem (Tri-TSP10N
3.5 for bi-objective tasks and 3 = 4.5 for tri-objective tasks.
29

---

Table 8: HV on large Bi-TSP instances.
Problem Size
WE-CA
POCCO-W
Bi-TSP300
0.7441
0.7458
Bi-TSP500
0.7476
0.7592
Bi-TSPI000
0.7186
0.7400
Table 9: HV on large MOKP instances.
Problem Size
WE-CA
POCCO-W
Bi-KP300
0.6000
0.5947
Bi-KP500
0,4244
0.5658
Bi-KP1000
0.2439
0.8408
Table 10: HV of different PL methods on Bi-TSP instances.
Problem Size
POCCO-W
DPO
Bi-TSP20
0.6275
0.6276
Bi-TSP50
0.6411
0,6400
Bi-TSPI00
0.7055
0.7030
Bi-TSPI50
0.7033
0,6998
Bi-TSP200
0.7371
0.7331
BI-T5P100
MOCVRP100
0.706
0.409
0.706
0.409
0.7055
0,7052
0.706
0.7054
0.408
0.4084
0.4080
0.705
0.408
0,4082
2
0.705
0.408
0.705
0.408
0.705
0.407
0.704
0.407
0.704
0.407
3.5
3
3.5
8
Bi-KP100
Tri-TSP100
b2602
0.3602
0.3603
0.3603
0.325
0.4885
0.4983
0.300
8,6980
0.275
0.4573
14578
0.250
3.5
3.5
4.5
B
8
Figure 5: Effectis of the 8.
30

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Effects of the number of CCO layers. The CCO block sits in the decoder, whose sequence length T grows with problem size. Each additional CCO layer. therefore, adds full set of gating and expert computations at every decoding step. Hence, stacking too many CCO layers can inflate the runtime disproportionately. Table 11 reports the trade-off. Using two CCO layers yields the highest HV on both Bi-TSPI00 and Bi-TSP200, but increases inference time by roughly 60% and 130 %, respectively. A third layer further slows inference while slightly reducing HV. We therefore adopt single-layer CCO the default, which preserves most of the performance gain while keeping computation modest. For larger instances or latency-sensitive applications, the one-layer setting offers favorable balance between solution quality and speed.
Table 11: Effects of the number of CCO block layers.
Bi-TSP100
Bi-TSP200
Method
HV
Time
HV
Time
POCCO-W
0.7055
36s
0.7371
2.4
2 CCO layers
0.7058
59
0.7379
5.6m
CCO layers
0.7049
85
0.7352
7.9 m
Effects of the number of experts. We have added an ablation study to analyze the impact of the number of experts Table 12. Specifically, POCCO-V W (5E) refers to the original model presented in the paper, which includes 4 feedforward (FF) experts and I parameter-free identity (ID) expert. We additionally evaluate four variants:
1. POCCO-W (3E): 2 FF + ID;
2. POCCO-W (9E): 8 FF + ID;
3. POCCO-W 2D): FF I ID, trained on twice the amount data;
4. POCCO-W (17E): FF + ID.
All variants are trained under the same settings as POCCO-W (5E) for a fair comparison, except for POCCO-W (9E_2D), which is trained on more data.
As shown in Table 12, all POCCO-W W variants outperform the backbone model WE-CA, confirming the effectiveness of the expert- based architecture. Moreover, POCCO-W (3E) and POCCO-W (5E) achieve better overall performance than POCCO-W -W (9E) and POCCO-W W (17E), the latter of which requires additional data scaling realize performance gains. To strike better balance between computational cost and solution quality, we select POCCO-W (5E) as our default model.
Table 12: Effects of the number of experts.
Problem Size
WE-CA
POCCO-W (3R)
POCCO-W (SE)
Exp. (9F)
Exp. (9F_ 2D)
Exp_num (17F)
Bi-TSP20
0.6275
0.6275
0.6275
0.6275
Bi-TSP50
0.6392
0.6410
0,6411
0.6408
0.6411
0.64U8
0,7034
0.7057
0.7055
0.7050
0.7053
0.7050
Ri-TSP150
0.7008
0.7035
0.7033
0.7026
0.7031
0.7027
Bo-TSP200
0.7346
0.7363
0.7371
0.7364
0.7370
0,7361
Effects of Top k. Fig.6 plots validation HV on Bi-TSPI00 for & 3,) Selecting two experts per token (k=2) converges fastest and attains the highest final HV. single expert (k=l) limits model capacity, leading to slower early progress and slightly lower plateau. Choosing three experts (k=3) increases computation relative to k=2 yet offers no benefit and even marginally reduces HV, likely because the additional expert dilutes specialization and weakens sparsity. We therefore adopt k=2 as the default, balancing performance and efficiency.
M Experimental Results of Scalarized Subproblems
To assess subproblem optimality, we report scalarized objective values for three representative weight vectors = (),0), (0.5, 0.5). (0, 1) on Bi-TSP100 (Table 13). We also compare against
single-objective solvers LKH and POMO, each tuned to the corresponding subproblem. Among
31

---

0.7050
0.7025
0.7000
0.6975
HV
0.6950
0.6925
Top k=1
0.6900
Top k=2
0.6875
Top k=3
1
20
40
60
80
120 Epoch
100
140
160
180
200
Figure Effects of Top
neural MOCOP methods, POCCO-W - attains the smallest optimality gaps across all weight settings. Ablating CCO (POCCO-Aug w/o CCO) increases optimality gaps across all settings, and eliminating preference learning as well (WE-CA-Aug) further degrades performance. Notably, POCCO-W-Aug outperforms POMO-Aug on the subproblem with À (0,
Table 13: Performance comparison under different weight settings (w1, W2).
À (1,0)
x (0.5, 0.5)
À (0,
Method
Obj
Gap
Obj
Gap
Obj
Gap
WS-LKH
7.7632
0.00%
17.3094
0.00%
7.7413
0.00%
POMO-Aug
7.7659
0.03%
17.4421
0.77%
7.7716
0.39%
WE-CA-Aug
7.8132
0.64%
17.4994
1.10%
7.7888
0.61%
POCCO-AUE w/o CCO
7.7970
0.44%
17.4629
0.89%
7.7772
0.46%
POCCO-
POCCO-W-Aug W-Aug
7.7827
0.25%
17.4460
0.79%
7.7602
0.24%
N
Comparison with Tchebycheff Scalarization
Intuitively, our POCCO framework is applicable to any MOCOP where the objectives can be scalarized using decomposition- techniques, including weighted-sum (WS), Tchehycheff (TCH), and penalty-based boundary intersection (PBI). This is because the preference signal between two solutions is determined solely by the ordering of their scalarized objective values, regardless of the specific scalarization method used.
To empirically demonstrate this generality, added an experiment comparing POCCO-W trained with WS and TCH scalarization methods. As shown in Table 14, WS consistently outperforms TCH on Bi-TSP50 to Bi-TSP200, indicating that WS offers more robust performance across different problem sizes. This observation is also consistent with findings in [6], which report that WS- despite its simplicity-often outperforms TCH in the studied MOCOP setting. general, however, WE- CA (TCH) performs worse than WE-CA (WS), which achieves score of 0.6392, and is further outperformed by POCCO-W (TCH), which achieves 0.6395 on Bi-TSP50.
GPU Memory Usage
We present comparison of GPU memory usage between POCCO-W (ours), the backbone model (WE-CA), and CNH in Table 15. All models are trained across problem instances with size 50. As shown in Table 15, POCCO-W incurs higher GPU memory usage, approximately doubling that of
32

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Table 14: HV of Different Decomposition Approaches on Bi-TSP Instances.
Problem Size
WE-CA (WS)
POCCO-W (WS)
POCCO-W (TCH)
Bi-TSP50
0.6392
0.6411
0.6395
Bi-TSPI00
0.7034
0.7055
0.7031
Bi-TSP150
0.7008
0.7033
0,6998
Bi-TSP200
0.7346
0.7371
0.7329
Table 15: Comparison of GPU Memory Usage.
Problem Size
WE-CA
CNH
POCCO-W
Bi-TSP50
1051 MB
1077 MB
1849 MB
MOCVRP50
1790 MB
1501 MB
4522 MB
Bi-KP50
916 MB
916 MB
2417 MB
Tri-TSP50
1052 MB
1077 MB
2383 MB
the baselines WE-CA and CNH. However, this increased resource cost is accompanied by significant improvements in solution quality, as demonstrated in our experimental results.
P Summary of Decomposition-based Neural MOCOP Solvers
Table 16 compares key features of decomposition- neural MOCOP solvers. POCCO, which establishes new SOTA performance, is a plug-and-plaa framework that augments any existing solver. inserts a CCO block that learns a diverse ensemble of policies, adding parameters yet surpassing prior methods. Crucially, each subproblem activates only two experts (one of which may be parameter-free identity expert), so the extra computational load remains minimal. POCCO also employs pairwise preference learning approach that further boosts performance without introducing additional parameters.
Table 16: Summary of the decomposition--asse neural MOCOP solvers.
Method
Learning method
Paradigm
#Parameters
DRL-MOA
Transfer learning+RL.
one-to-one
133.37M
MDRL.
Meta leL
one-to-one
133.37M
EMNH
Meta learning+RL.
one-to-one
133.37M
PMOCO
RL.
one-to-many
1.50M
CNH
RL
one-to-many
1.63M
POCCO-C
Preference learning
one-to-many
2.16M
WE-CA
RL
one-to-many
1.47M
POCCO-W
Preference learning
one-to-many
2.00M
Broader Impacts
POCCO offers several positive societal impacts. By dynamically routing computation and learning from preference signals, it accelerates multi-objective decision-making in logistics, manufacturing, and energy planning, reducing costly trial-and-error loops and boosting operational efficiency. Its conditinnalcoomptation design activates only the required network capacity for each subproblem, cutting FLOP counts and energy consumption relative to dense models of comparable accuracy. Finally, by encouraging exploration and adaptive capacity allocation, POCCO broadens the applica- bility of neural solvers in complex optimization tasks, advancing both Al and operations research and enabling practitioners tackle larger, real-world problems with fewer computational resources.
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R
Licenses for Existing Assets
The used assets in this work are listed in Table 17. Where applicable, we reference publicly available implementations for evaluation or reproduction purposes. Our source code is released under the MIT License.
Table 17: Used assets, licenses, and their usage.
Type
Asset
License
Usage
LKH [20]
Available for academic
Evaluation
DRL-MOA [34]
No license (assumed all rights reserved)
Evaluation
Code
MDRL [67]
No license (assumed all rights reserved)
Evaluation
EMNH [7]
No license (assumed all rights reserved)
Evaluation
PMOCO [38]
MIT License
Evaluation
CNH [14]
MIT License
Revision
WE-CA [6]
MIT License
Revision
Datasets
Chen et al.[6]
MIT License
Evaluation
34


Reviews:
[{"review_id": "pICNB3Z2yW", "reviewer": "Reviewer_ETVh", "summary": "Current methods for multi-objective combinatorial optimization problems (MOCOPs) often rely on a single neural network to handle all subproblems with different weight vectors, resulting in a limited model capacity for diverse subproblems. On the other hand, current methods usually use REINFORCE and use scalarized objective values as the reward signals, introducing high variance. To address these issues, the paper introduces a novel method called POCCO, which increases model capacity by conditional computational (CCO) block and utilize preference-based reward for a better convergence. Experiments across several classic MOCOP benchmarks show that POCCO outperforms all the learning-based methods, with an increased but still acceptable computational time cost.", "questions": "- Could you provide a comparison on the resource usage (e.g., GPU memory usage) of the methods? This would provide a more complete understanding of the practical trade-offs.\n- Can POCCO be used in the tasks where preference among the objective cannot be presented as a linear weighted sum, but only a more complex combination (e.g. Tchebycheff scalarization)?\n\nI am willing to change my rating if all the concerns and questions is answered or addressed.", "limitations": "The CCO block may require more computational cost (including time and resource usage) compared to other learned-based methods.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n- The paper introduces both a new architecture and a new training method for MOCOPs, which are conceptually sound and well-motivated for MOCOPs.\n- In the experiment, the proposed method POCCO outperforms all the learning-based methods in terms of hyper-volume and gap. The time usage is longer but acceptable. The experiment is generally adequate.\n- The paper is clearly written, well-organized, and easy to follow.\n\nWeaknesses\n\n- The CCO block may introduce more computational overhead (including time and resource usage) compared to other learned-based methods. The longer computational time is evident in the reported experimental results.\n- The influence of the number of experts is not analyzed. It might be critical to the performance.\n- The source code is not yet available in the supplementary material, which may hinder the evaluation on reproducibility.", "was_revised": true, "final_justification": "The authors have addressed most of my concerns. I am planning to keep my rating unchanged.", "rebuttal": "We genuinely value and appreciate the time and effort the reviewer dedicated to evaluating our paper and offering insightful feedback. Here are our detailed responses to your comments, where W denotes Weakness (W1-W3) and Q denotes Question (Q1-Q2).\n\n**W1: The CCO block may introduce more computational overhead (including time and resource usage) compared to other learned-based methods.**\n\nWe acknowledge that the CCO block introduces additional computational overhead due to the extra parameters added on top of the backbone model. While this results in a moderate increase in runtime (as shown in Tables 1-2) and resource usage (as noted in our response to `Q1`), the improvement in solution quality is substantial. As demonstrated in our ablation study, the CCO block plays a key role in achieving these performance gains, justifying its inclusion in the overall design. We believe this trade-off between efficiency and performance is reasonable, especially considering the gains over other learning-based methods.\n\n**W2: The influence of the number of experts is not analyzed.**\n\nThanks for your comment. We have added an ablation study to analyze the impact of the number of experts in Table A. Specifically, POCCO-W (5E) refers to the original model presented in the paper, which includes 4 feedforward (FF) experts and 1 parameter-free identity (ID) expert. We additionally evaluate four variants:\n* POCCO-W (3E): 2 FF + 1 ID;\n* POCCO-W (9E): 8 FF + 1 ID;\n* POCCO-W (9E_2D): 8 FF + 1 ID, trained on twice the amount of data;\n* POCCO-W (17E): 16 FF + 1 ID.\n\nAll variants are trained under the same settings as POCCO-W (5E) for a fair comparison, except for POCCO-W (9E_2D), which is trained on more data. \n\nAs shown in Table A, all POCCO-W variants outperform the backbone model WE-CA, confirming the effectiveness of the expert-based architecture. Moreover, POCCO-W (3E) and POCCO-W (5E) achieve better overall performance than POCCO-W (9E) and POCCO-W (17E), the latter of which require additional data scaling to realize performance gains. To strike a better balance between computational cost and solution quality, we select POCCO-W (5E) as our default model.\n\n**Table A: HV on Bi-TSP Instances.**\n| Problem Size | WE-CA | POCCO-W (3E)| POCCO-W (5E)  | Exp_num (9E) |Exp_num (9E_2D) |Exp_num (17E) |\n|:-------------|:--------:|:-----------------:|:------------------------:|:-----------------:|:-----------------:|:-----------------:|\n| Bi-TSP20  | 0.6270 | **0.6275** | **0.6275** | **0.6275** |**0.6275**|**0.6275** |\n| Bi-TSP50  | 0.6392 | 0.6410 | **0.6411** | 0.6408 | **0.6411** | 0.6408 |\n| Bi-TSP100 | 0.7034 | **0.7057** | 0.7055 | 0.7050 |   0.7053   | 0.7050 |\n| Bi-TSP150 | 0.7008 | **0.7035** | 0.7033 | 0.7026 |   0.7031 | 0.7027 |\n| Bo-TSP200 | 0.7346 | 0.7363 | **0.7371** | 0.7364 |   0.7370| 0.7361 |\n\n\n**W3: The source code is not yet available.**\n\nWe are committed to publicly releasing our code and dataset upon acceptance. Due to NeurIPS 2025 rebuttal policy, which prohibits the inclusion of external links, we were unable to share our source code via an anonymous repository. However, we will ensure full code and data availability to support reproducibility once the paper is published.\n\n\n**Q1: Could you provide a comparison on the resource usage (e.g., GPU memory usage) of the methods? This would provide a more complete understanding of the practical trade-offs.**\n\nFollowing the reviewer's suggestion, we present a comparison of GPU memory usage between POCCO-W (ours), the backbone model (WE-CA), and CNH in Table B. All models are trained as unified models across problem instances with size $n =50$. As shown in Table B, POCCO-W incurs higher GPU memory usage, approximately doubling that of the baselines WE-CA and CNH. However, this increased resource cost is accompanied by significant improvements in solution quality, as demonstrated in our experimental results. We will include a more detailed comparison in the final version of the paper to provide a clearer understanding of the trade-offs involved.\n\n**Table B: Comparison of GPU Memory Usage.**\n\n| Problem Size| WE-CA |CNH |POCCO-W |\n|-----|-----|-----|-----|\n| Bi-TSP50  |  1051 MB |1077 MB | 1849 MB|\n| MOCVRP50  |  1790 MB |1501 MB | 4522 MB|\n| Bi-KP50  |  916 MB |916 MB | 2417 MB|\n| Tri-TSP50  |  1052 MB |1077 MB|  2383 MB|\n\n\n**Q2: Can POCCO be used in the tasks where preference among the objective cannot be presented as a linear weighted sum, but only a more complex combination (e.g. Tchebycheff scalarization)?**\n\nIntuitively, our POCCO framework is applicable to any MOCOP where the objectives can be scalarized using decomposition-based techniques, including weighted-sum (WS), Tchebycheff (TCH), and penalty-based boundary intersection (PBI). This is because the preference signal between two solutions is determined solely by the ordering of their scalarized objective values, regardless of the specific scalarization method used.\n\nTo empirically demonstrate this generality, we added an experiment comparing POCCO-W trained with WS and TCH scalarization methods. As shown in Table C, WS consistently outperforms TCH on Bi-TSP50 to Bi-TSP200, indicating that WS offers more robust performance across different problem sizes. This observation is also consistent with findings in [1], which report that WS—despite its simplicity—often outperforms TCH in the studied MOCOP setting. Due to the limited rebuttal period, we will include the detailed results for WE-CA (TCH) in the revised paper. In general, however, WE-CA (TCH) performs worse than WE-CA (WS), which achieves a score of 0.6392, and is further outperformed by POCCO-W (TCH), which achieves 0.6395 on Bi-TSP50.\n\n**Table C: HV of Different Decomposition Approaches on Bi-TSP Instances.**\n| Problem Size | WE-CA (WS) | POCCO-W (WS) | POCCO-W (TCH) |\n| :----------: | :--------: | :----------: | :-----------: |\n|   Bi-TSP50   |   0.6392   |    0.6411    |    0.6395     |\n|  Bi-TSP100   |   0.7034   |    0.7055    |    0.7031     |\n|  Bi-TSP150   |   0.7008   |    0.7033    |    0.6998     |\n|  Bi-TSP200   |   0.7346   |    0.7371    |    0.7329     |\n\n[1] Rethinking neural multi-objective combinatorial optimization via neat weight embedding, in ICLR 2025."}, {"review_id": "hMnq7vHjYx", "reviewer": "Reviewer_ZHcK", "summary": "The paper introduces POCCO (Preference-driven multi-Objective Combinatorial optimization with Conditional COmputation), a framework that can be wrapped around existing neural solvers for multi-objective combinatorial optimization problems (MOCOPs).\nIn particular, POCCO adds two orthogonal components:\n1. Conditional Computation (CCO block): a sparse MoE / Mixture-of-Depth module inserted in the decoder that routes each scalarised sub-problem to a small, specialized subset of experts (or an identity path), efficiently scaling model capacity.\n2. Preference-driven Learning (PL): Instead of REINFORCE with scalar rewards, the policy is trained on pairwise “win/lose” comparisons via a Bradley-Terry likelihood, using normalized log-likelihood as an implicit reward.\n\nIntegrated into two SOTA baselines (CNH and WE-CA), the resulting POCCO-C and POCCO-W outperform all neural and classical baselines on four benchmarks (Bi-/Tri-TSP, Bi-CVRP, Bi-KP) and show better OOD size generalization. The authors also discuss remaining limitations, such as scaling to real-world constraints and very large instances.", "questions": "## Questions\n\n1. How does performance vary with Top-k and the number of experts? Could very large expert pools cause over-fragmentation or collapse?\n2. Have you compared BT-likelihood with more recent objectives such as DPO?\n3. How would CCO interact with the constraint-handling modules suggested for future work? Could routing specialise not just by weight vector but by constraint pattern?\n\n## Possible typos\n\n- l.169-170 – “it combines of a mixture-of-experts…” → “combines a mixture-of-experts”", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "## Strengths\n\n- Novelty: Simple, general mechanisms that can be “plugged in” to many existing models.\n- Experiments: 4 benchmarks × 3 sizes, 3 metric types, ablations on CCO variants and hyper-parameters, OOD tests.\n- Training efficiency: PL halves convergence time vs. REINFORCE.\n- Generalizability: Single model handles multiple sizes, beats multi-model baselines, and generalises to KroAB100-200 instances.\n- Paper clarity: Background and motivation sections relate CCO + PL to known MoE and preference-learning literature.\n\n## Weaknesses\n\n- Theoretical grounding: No formal analysis of convergence or expressivity. All results rely solely on empirical evidence (the authors acknowledge no new theory).\n- Experimental scope: Instances top out at 100 nodes for TSP/CVRP or 200 items for KP. Real-world constraints are left to future work.\n- Missing ablations: CCO adds parameters, yet the paper does not quantify the overhead of expert routing.\n- Writing: Several minor typos and grammatical errors that slightly hinder readability.", "was_revised": true, "final_justification": "I have updated my original score following the provided theoretical grounding and formulation of the PL objective and beginning of gradient variance analysis.", "rebuttal": "Thank you for your insightful comments.\n\n**W1: Lacks theoretical analysis on convergence or expressivity.**\n\nDue to character limits, please refer to our response to Reviewer CeFA–W1 for details on W1.\n\n**W2: Limited problem scale; real-world constraints not addressed.**\n\nWe mainly follow prior work [1-3], which primarily evaluates methods on MOTSP/MOCVRP with 20/50/100 nodes and MOKP with 50/100/200 items.\n\nTo evaluate the out-of-distribution generalization ability of our method, we report additional results on larger Bi-TSP instances (150 and 200 nodes) as well as benchmark instances (KroAB150 and KroAB200) in Appendix H and I, respectively. Furthermore, to demonstrate scalability, we include new experimental results on even larger problems including Bi-TSP with 300/500/1000 nodes and MOKP with 300/500/1000 items. These results are summarized in Table D and Table E. As shown, POCCO-W consistently outperforms the baseline WE-CA in most large-scale cases.\n\nFollowing the reviewer’s suggestion to address the applicability of our method under more complex constraints, we conducted an additional experiment on the multi-objective capacitated vehicle routing problem with time windows (MOCVRPTW), as summarized in Table F. In the MOCVRPTW, each customer node $v_i$ is associated with a time window $[e_i, l_i]$ and a service time $s_i$[4]. A vehicle must start serving customer $v_i$ in the time slot from $e_i$ to $l_i$.  If the vehicle arrives earlier than $e_i$, it has to wait until $e_i$. All vehicles must return to the depot $v_0$ no later than $l_0$. This study considers three objectives: minimizing the total travel distance, minimizing the number of routes, and minimizing the average route length per customer served. As shown in Table F, POCCO-W consistently outperforms WE-CA on MOCVRPTW tasks as well.\n\n**Table E: HV on large Bi-TSP instances.**\n| Problem Size | WE-CA HV | POCCO-W HV |\n|--------------|:--------:|:----------:|\n| Bi-TSP300     | 0.7441   | **0.7458**     |\n| Bi-TSP500     | 0.7476   | **0.7592**     |\n| Bi-TSP1000    | 0.7186   | **0.7400**     |\n\n**Table F: HV on large MOKP instances.**\n| Problem Size | WE-CA HV | POCCO-W HV |\n|--------------|:--------:|:----------:|\n| Bi-KP300      | **0.6000**   | 0.5947     |\n| Bi-KP500      | 0.4244   | **0.5658**     |\n| Bi-KP1000     | 0.2439   | **0.8408**     |\n\n**Table G: HV on MOCVRPTW instances.**\n| Problem Size | WE-CA HV | POCCO-W HV |\n|--------------|:--------:|:----------:|\n| MOCVRPTW20    | 0.5465   | **0.5496** |\n| MOCVRPTW50    | 0.5499   | **0.5576** |\n| MOCVRPTW100   | 0.5331   | **0.5375** |\n\n**W3: Missing ablation on CCO’s routing overhead.**\n\nWe have quantified the overhead introduced by the CCO block in Table 8 (Appendix L), which compares the total number of parameters across various neural MOCOP solvers. Our CCO block, designed to learn a diverse ensemble of policies, adds additional parameters yet achieves superior performance over prior methods. Specifically, the additional parameters in POCCO-W primarily come from four FF experts (totaling 526,848 parameters) and a lightweight router module (1,280 parameters). It is worth noting that during inference, each subproblem activates only two experts, one of which may be the parameter-free ID expert. This design helps keep the computational overhead manageable.\n\nTo further address your concern, we conducted an additional set of experiments comparing several neural MOCOP solvers in terms of model size, training time, GPU memory usage (on Bi-TSP50), and solution quality (HV). As shown in Table H, while POCCO-W has moderately higher overhead than CNH and WE-CA, it achieves better solution quality. We believe this represents a reasonable trade-off between model complexity and performance.\n\n**Table H: Overhead of Different Neural MOCOP solvers.**\n| Method    | Parameters | Training Time| GPU Usage (Bi-TSP50)|HV (Bi-TSP50)|\n|-----------|:-----------:|:-------------:|:-----------:|:-----------:|\n| CNH       | 1.63M       | 17h |1077 MB | 0.6387 |\n WE-CA     | 1.47M       | 11h |1051 MB | 0.6392 | \n| POCCO-W   | 2.00M   | 36h |1849 MB  | 0.6411 | \n\n**W4: Minor writing issues.**\n\nWe acknowledge minor typos and grammatical issues in the manuscript and will thoroughly revise the paper to improve clarity and readability.\n\n**Q1:  How do Top-k and expert count affect performance? Risk of collapse?**\n\nWe analyzed the impact of Top-k selection on both training dynamics and final performance in Figure 6 (Appendix J). POCCO-W with Top-k = 2 achieves the fastest convergence and the highest final HV compared to Top-k = 1 and Top-k = 3. To further support this observation, Table I presents a detailed ablation study, where POCCO-W (Top-k = 2) consistently outperforms other settings across all tested problem sizes.\n\nWe have added an ablation study to analyze the impact of the number of experts in Table A (due to character limits, please refer to our response to Reviewer ETVh–W2 for details on Table A). Specifically, POCCO-W (5E) refers to the original model presented in the paper, which includes 4 feedforward (FF) experts and 1 parameter-free identity (ID) expert. We additionally evaluate four variants:\n* POCCO-W (3E): 2 FF + 1 ID;\n* POCCO-W (9E): 8 FF + 1 ID;\n* POCCO-W (9E_2D): 8 FF + 1 ID, trained on twice the amount of data;\n* POCCO-W (17E): 16 FF + 1 ID.\n\nAll variants are trained under the same settings as POCCO-W (5E) for a fair comparison, except for POCCO-W (9E_2D), which is trained on more data. \n\nAs shown in Table A, all POCCO-W variants outperform the backbone model WE-CA, confirming the effectiveness of the expert-based architecture. Moreover, POCCO-W (3E) and POCCO-W (5E) achieve better overall performance than POCCO-W (9E) and POCCO-W (17E), the latter of which require additional data scaling to realize performance gains. To strike a better balance between computational cost and solution quality, we select POCCO-W (5E) as our default model.\n\nDue to computational resource constraints and rebuttal timeframe limitations, we did not test even larger expert pools (e.g., 25 or 49 experts). Nevertheless, POCCO-W (17E) still performs robustly, showing no collapse. We will further explore and provide deeper analyses with larger expert pools in the final manuscript.\n\n**Table I: HV of dfferent Top-k on Bi-TSP instances.**\n| Problem Size | Top-k = 1 | Top-k = 2 | Top-k = 3 |\n|:------------:|:---------:|:---------:|:---------:|\n| Bi-TSP20  | 0.6274 | **0.6275** | **0.6275** |\n| Bi-TSP50  | 0.6404 | **0.6411** | 0.6409 |\n| Bi-TSP100 | 0.7046 | **0.7055** | 0.7050 |\n| Bi-TSP150 | 0.7020 | **0.7033** | 0.7025 |\n| Bi-TSP200 | 0.7354 | **0.7371** | 0.7364 |\n\n**Q2: Comparison with DPO?**\n\nWe have added a comparative experiment between our preference learning (PL) method and the recent DPO objective, as shown in Table H. Since DPO requires two models, i.e., a policy model and a reference model, we use the same architecture and initialization for both, with the reference model updated from the previous epoch. This setup significantly increases training time for DPO (122 h vs. 36 h for POCCO-W).\n\nAs shown in Table H, while DPO achieves slightly better performance on the smallest instance (MOTSP20), it is consistently outperformed by POCCO-W on larger instances (MOTSP50–200). These results suggest that our PL method offers a more favorable trade-off in terms of both training efficiency and solution quality, especially for large-scale problems.\n\n**Table J: HV of different PL methods on Bi-TSP instances.**\n| Problem Size | POCCO-W  | DPO   |\n|--------------|:----------:|:-------:|\n| Bi-TSP20      | 0.6275     | **0.6276**  |\n| Bi-TSP50      | **0.6411**     | 0.6400  |\n| Bi-TSP100     | **0.7055**     | 0.7030  |\n| Bi-TSP150     | **0.7033**     | 0.6998  |\n| Bi-TSP200     | **0.7371**     | 0.7331  |\n\n**Q3: Can CCO routing adapt to constraint patterns?**\n\nOur CCO module between the decoder’s MHA and the compatibility layer is orthogonal to the constraint-handling mechanism, which is primarily realized via masking schemes in the MHA and compatibility modules. In future work, we envision two possible directions where CCO could interact with constraint-handling modules:\n* **Single-task with hard constraints:** MOCOPs with hard constraints (e.g., TSPTW) are challenging, as identifying valid masking patterns is itself an NP-hard problem. In such cases, we could use an auxiliary decoder to learn to predict feasible masking patterns [5], while the main decoder with the CCO learns the solution policy. Through joint training or mutual distillation, the expert routing in CCO can specialize not just by weight vector, but also implicitly by constraint structure. This interaction encourages experts to develop sub-policies aligned with recurring constraint motifs.\n* **Multi-task generalization with varying constraint patterns:** For multi-task settings where the model needs to generalize across multiple MOCOPs with heterogeneous constraints (e.g., TSP with capacity, VRP with time windows, KP with budget limits), we can extend the input to the CCO router by concatenating a binary constraint indicator vector that encodes the presence or type of constraints for each task [4]. This approach allows experts to specialize in sub-distributions of tasks sharing similar constraint structures.\n\nIn both settings, the CCO module functions as a flexible capacity allocator within the decoder, and its interaction with constraint-handling modules can be enhanced either implicitly or explicitly.\n\n[1] Pareto set learning for neural multi-objective combinatorial optimization, in ICLR 2022.\n\n[2] Conditional neural heuristic for multiobjective vehicle routing problems, TNNLS, 2024.\n\n[3] Rethinking neural multi-objective combinatorial optimization via neat weight embedding, in ICLR 2025.\n\n[4] MVMoE: Multi‑Task Vehicle Routing Solver with Mixture‑of‑Experts, in ICML 2024.\n\n[5] Learning to handle complex constraints for vehicle routing problems, in NeurIPS 2024."}, {"review_id": "Qs0jqSEQrB", "reviewer": "Reviewer_CeFA", "summary": "This paper proposes POCCO, a plug-and-play framework for Multi-Objective Combinatorial Optimization Problems (MOCOPs) using deep reinforcement learning. Unlike existing methods that rely on scalarized rewards and shared models across subproblems, POCCO introduces two key components: a conditional computation block that routes subproblems to specialized neural architectures, and a preference-driven optimization algorithm that leverages pairwise preferences.", "questions": "1. How is the routing mechanism in the CCO block learned? Specifically, how does the sparse gating network select between FF and ID experts, and what role does the Top-k operation play?\n\n2. When sampled solutions are non-dominated, how does the scalarized value determine the winner in preference learning? Could this bias the Pareto front exploration?\n\n3. The paper claims efficiency gains—could you break down the contributions of the CCO block and preference-driven training to the overall speedup?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. POCCO introduces a novel plug-and-play framework that effectively tackles key limitations of existing MOCOP methods, including insufficient exploration and limited model capacity. \n2. The paper provides substantial experimental verification across multiple classic MOCOP benchmarks, exhibiting strong generalization capabilities and excellent performance even when faced with unseen problem scales and out-of-distribution instances. \n\nWeaknesses:\n1. The paper lacks theoretical support for its design choices. For instance, it’s unclear how the CCO block enhances exploration or representation learning, or how preference learning mitigates gradient variance in REINFORCE.\n2. When generating preference pairs, the paper samples two candidate solutions from the policy and determines superiority based on the scalarized objective values. However, in complicated multi-objective settings, solutions can be mutually non-dominated.", "was_revised": true, "final_justification": "The authors have addressed all my concerns.", "rebuttal": "We appreciate your valuable comments.\n\n**W1: Lacks theoretical justification how CCO aids representation and how PL reduces gradient variance.**\n\nThe majority of the work [1–5] within the neural multi-objective combinatorial optimization (MOCO) literature focused on the empirical side, making the provision of a solid theoretical analysis within the tight rebuttal period a non-trivial challenge. Here, we try to give an in-depth empirical analysis concerning the effects of the CCO module and preference learning method, with the aim of offering valuable insights for future research.\n\nThe CCO block, positioned between the decoder’s attention and compatibility layers, enhances the model's representation capacity by dynamically selecting among multiple experts for each subproblem. The model can conditionally adapt its decoding behavior based on both context and objective preference. This modularity encourages expert specialization, enabling the decoder to capture a richer and more diverse set of mappings from inputs to actions. Since different weight vectors (in decomposition-based learning) induce different subproblem characteristics, the CCO router can learn to route similar subproblems to similar experts, effectively partitioning the representation space [6-7]. This facilitates both exploration across diverse sub-tasks and robustness to task shift, which is particularly beneficial in multi-objective optimization. \n\nCompared to standard REINFORCE-based reinforcement learning, our preference learning (PL) approach offers significant practical advantages. Instead of relying on high-variance reward signals (as in REINFORCE), PL compares solutions under the same weight vector, providing a low-variance signal. We provide concrete evidence of this in Table D, which reports the average gradient variance in the first five batches of training POCCO-W under both RL and PL frameworks. Across three MOTSP settings, PL consistently exhibits two to four orders of magnitude lower variance, leading to faster convergence and more stable optimization dynamics.\n\n**Table D: Gradient variance in REINFORCE vs. preference learning.** \n| Problem Size | Batch | RL Variance | PL Variance |\n|--------------|:-----:|:----------------:|:----------------:|\n| MOTSP20      |   1   |     0.054648     |     0.000314     |\n|              |   2   |     0.039951     |     0.000252     |\n|              |   3   |     0.019038     |     0.000191     |\n|              |   4   |     0.009093     |     0.000115     |\n|              |   5   |     0.010742     |     0.000104     |\n| MOTSP50      |   1   |     0.474784     |     0.000142     |\n|              |   2   |     0.220530     |     0.000077     |\n|              |   3   |     0.140275     |     0.000048     |\n|              |   4   |     0.092114     |     0.000040     |\n|              |   5   |     0.065286     |     0.000031     |\n| MOTSP100     |   1   |    10.124832     |     0.000059     |\n|              |   2   |     6.355499     |     0.000039     |\n|              |   3   |     3.461700     |     0.000032     |\n|              |   4   |     1.755804     |     0.000021     |\n|              |   5   |     1.111246     |     0.000014     |\n\n\n\n**W2 & Q2: How does PL handle non-dominated pairs, and could bias Pareto exploration?**\n\nOur method adopts a decomposition-based framework, where the MOCOP is divided into a set of scalarized subproblems, each associated with a weight vector. During training, the goal is to find high-quality solutions with respect to each subproblem, rather than globally exploring the entire Pareto front at once. In this context, preference learning operates at the subproblem level. We acknowledge, however, that sampled solutions may occasionally be mutually non-dominated in the original objective space. To investigate the effect of this scenario, we conducted an additional experiment where preference signals for such non-dominated pairs were set to 0, effectively treating them as equivalent. The results, shown in Table K, indicate that this modification (NDS) significantly degrades model performance across all Bi-TSP instances.\n\nThis observation suggests that introducing indifference signals for non-dominated pairs hinders the learning process. One reason is that it reduces the amount of effective preference supervision per subproblem, especially in early training stages when many solutions are of similar quality. In contrast, using scalarized values provides a consistent and differentiable training signal aligned with the decomposition objective. Therefore, although scalarized comparisons may not reflect global Pareto dominance in every case, they remain well-justified and necessary within the decomposition-based learning paradigm, and do not appear to bias the overall Pareto front approximation based on our empirical findings.\n\n**Table K: Perfomance on Bi-TSP instances when neutralizing non-dominated pairs.**\n| Problem Size | POCCO-W HV | NDS HV |\n|--------------|:----------:|:------:|\n| Bi-TSP20      | 0.6275     | 0.5516 |\n| Bi-TSP50      | 0.6411     | 0.5160 |\n| Bi-TSP100     | 0.7055     | 0.5601 |\n| Bi-TSP150     | 0.7033     | 0.5494 |\n| Bi-TSP200     | 0.7371     | 0.5809 |\n\n**Q1: How is CCO routing learned, and what role does the Top-k play?**\n\nWe have provided an explanation of the routing mechanism in Appendix C.3 and elaborate on it below. \n\nOur CCO block employs a subproblem-level gating mechanism to dynamically route each context vector $h_c$ to a subset of experts from a pool consisting of 4 FF experts and 1 ID expert (thus $m=4$). Let $d$ be the hidden dimension and $W_G \\in \\mathbb{R}^{d \\times (m+1)}$ denote the trainable gating weight matrix. Given a batch of context vectors $X = \\left[ h_c^b \\right]_{b=1}^B \\in \\mathbb{R}^{B \\times d}$, where $B$ is the batch size, the gating network computes a score matrix: $H = X \\cdot W_G \\in \\mathbb{R}^{B \\times (m+1)}$, where  each element $H_b^j$ represents the affinity or preference (score) of subproblem b towards expert $j$.\n\nFor each subproblem, the router selects the Top-k highest-scoring experts from the score vector $H_b^j$.  These selected experts are then activated and their outputs are weighted by a softmax-normalized version of their scores. In our POCCO with $k = 2$, each subproblem is routed to its two highest-scoring experts. \n\nThis Top-k routing plays a critical role by reducing computation, as only $k$ experts are activated per subproblem (i.e., sparse activation widely used in MoE structure [8]). It also encourages expert specialization, since different subproblems (defined by context and weight vectors) tend to activate different subsets of experts. Additionally, it allows the inclusion of a parameter-free ID expert, which is often selected when minimal transformation is needed. This enables the router to learn when it is appropriate to leave a representation unchanged.\n\nWe analyzed the impact of Top-k selection on both training dynamics and final performance in Figure 6 (Appendix J). POCCO-W with Top-k = 2 achieves the fastest convergence and the highest final HV compared to Top-k = 1 and Top-k = 3. To further support this observation, Table I presents a detailed ablation study, where POCCO-W (Top-k = 2) consistently outperforms other settings across all tested problem sizes.\n\n**Table I: HV of dfferent Top-k on Bi-TSP instances.**\n| Problem Size | Top-k = 1 | Top-k = 2 | Top-k = 3 |\n|:------------:|:---------:|:---------:|:---------:|\n| Bi-TSP20  | 0.6274 | **0.6275** | **0.6275** |\n| Bi-TSP50  | 0.6404 | **0.6411** | 0.6409 |\n| Bi-TSP100 | 0.7046 | **0.7055** | 0.7050 |\n| Bi-TSP150 | 0.7020 | **0.7033** | 0.7025 |\n| Bi-TSP200 | 0.7354 | **0.7371** | 0.7364 |\n\n\n**Q3: What are the contributions of the CCO and PL to the overall speedup?**\n\nWe would like to clarify that the proposed CCO block is not designed to improve computational efficiency, rather to enhance the model’s representational flexibility. In fact, as it introduces additional parameters through the router and FF experts, it incurs moderate overhead in both model size and runtime, as quantified in Tables 1-2 and Table 8.\n\nThe efficiency gain we refer to in the paper primarily stems from our preference-driven training strategy and specifically refers to the model’s fast convergence during training. As shown in Figure 4(a) and Table L, our method achieves nearly 2× faster convergence compared to RL-based baselines under the same training conditions. This acceleration reduces the total wall-clock time required for training to reach high-quality solutions, which we consider a practical efficiency benefit during the learning phase.\n\nWe acknowledge that the current wording in the paper may have caused confusion by implying overall computational efficiency. We are committed to eliminating any ambiguity in the final version.\n\n**Table L: HV of Different Training Algorithm at Each Epoch**\n| Method      | Epoch 1 | Epoch 50 | Epoch 100 | Epoch 150 | Epoch 200 |\n|-------------|:-------:|:-------:|:--------:|:--------:|:--------:|\n| POCCO-W (RL)| 0.6838  | 0.7009  | 0.7023   | 0.7032   | 0.7046   |\n| POCCO-W (PL)| 0.6884  | 0.7017  | 0.7037   | 0.7043   | 0.7055   |\n\n[1] Pareto set learning for neural multi-objective combinatorial optimization, in ICLR 2022.\n\n[2] Neural multi-objective combinatorial optimization with diversity enhancement, in NeurIPS 2023\n\n[3] Conditional neural heuristic for multiobjective vehicle routing problems, TNNLS, 2024.\n\n[4] Collaborative deep reinforcement learning for solving multi-objective vehicle routing problems, in AAMAS 2024.\n\n[5] Rethinking neural multi-objective combinatorial optimization via neat weight embedding, in ICLR 2025.\n\n[6] Towards understanding the mixture-of-experts layer in deep learning, in NeurIPS, 2022.\n\n[7] Statistical perspective of top-k sparse softmax gating mixture of experts, in ICLR 2024.\n\n[8] MVMoE: Multi‑Task Vehicle Routing Solver with Mixture‑of‑Experts, in ICML 2024."}, {"review_id": "XpmxPA9OeZ", "reviewer": "Reviewer_kHSo", "summary": "This paper proposes POCCO, a preference-driven multi-objective combinatorial optimization framework with conditional computation, which achieves performance improvements on MOCOPs. While the approach is innovative and empirically validated, several issues remain clarification.", "questions": "- Does the preference learning approach offer theoretical guarantees on better convergence or exploration efficiency compared to traditional REINFORCE?\n- Are the experimental results still valid for larger-scale problem instances? What are the primary factors contributing to this limitation\n- Does the introduction of multiple FF experts and dynamic routing significantly increase computational cost? Considering the potential need to handle more objectives, how do the authors envision scaling POCCO efficiently to accommodate high-dimensional objective spaces?\n- Can the process of generating preference pairs introduce bias?\nThe paper reports strong out-of-distribution generalization performance. Is the claimed generalization ability dependent on the training distribution?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The proposed approach is technically sound.\n- The paper introduces a CCO in the decoder that dynamically routes subproblems through specialized feed-forward or identity experts based on context. \n- The paper proposes a preference-driven optimization algorithm inspired by the Bradley–Terry model, which learns from pairwise solution comparisons rather than absolute rewards. \n- The proposed POCCO is evaluated on several multi-objective combinatorial optimization benchmarks.\n\n\nWeaknesses:\n- While related work is mentioned in the introduction and appendix, the paper lacks a dedicated literature review section. This makes it difficult to understand the development trajectory of existing MOCOP methods and POCCO’s positioning within the field.\n- The model complexity and computational cost are not clearly addressed, especially with the inclusion of multiple feed-forward experts and dynamic routing. Further clarification on training and inference efficiency—particularly in terms of parameter count, runtime, and scalability—is needed.\n- The concept of “Preference Learning” is vaguely defined.", "was_revised": true, "final_justification": "The authors have addressed my concerns.", "rebuttal": "We greatly appreciate the reviewer for providing constructive comments. Here are our detailed responses to your comments, where W denotes Weakness (W1-W3) and Q denotes Question (Q1-Q4).\n\n**W1: Lacks a dedicated related work section.**\n\nWe acknowledge the missing related work section and will move relevant content from the appendix into the main text. A comprehensive literature review will be added in the final version to clarify the development of MOCOP methods and POCCO’s position.\n\n**W2 & Q3: Clarify training/inference efficiency and scalability. Does CCO add significant cost, and how does POCCO scale to many-objective problems?**\n\nWe conducted an additional set of experiments comparing several neural MOCOP solvers in terms of model size, training time, GPU memory usage (on Bi-TSP50), and solution quality (HV). As shown in Tables 1–2 and Table H, POCCO-W incurs moderately higher training and inference overhead than CNH and WE-CA, but consistently achieves better performance. We believe this represents a reasonable trade-off between model complexity and performance.\n\nSpecifically, the additional parameters in POCCO-W primarily come from four FF experts (totaling 526,848 parameters) and a lightweight router module (1,280 parameters). It is worth noting that during inference, each subproblem activates only two experts, one of which may be the parameter-free ID expert. This design helps keep the computational overhead manageable.\n\n\n**Table H: Overhead of Different Neural MOCOP solvers.**\n| Method    | Parameters | Training Time| GPU Usage (Bi-TSP50)|HV (Bi-TSP50)|\n|-----------|:-----------:|:-------------:|:-----------:|:-----------:|\n| CNH       | 1.63M       | 17h |1077 MB | 0.6387 |\n WE-CA     | 1.47M       | 11h |1051 MB | 0.6392 | \n| POCCO-W   | 2.00M       | 36h |1849 MB  | 0.6411 | \n\nAs for scalability to high-dimensional objective spaces, POCCO is designed within a decomposition-based framework where each training subproblem is defined by a scalarization (e.g., weighted-sum or Tchebycheff). This means that the model does not directly operate in the full objective space, but instead solves scalarized subproblems, allowing it to scale independently of the number of objectives.\n\nNevertheless, we recognize that higher-dimensional Pareto fronts may require denser sampling of preference vectors, which could increase training cost. To address this, POCCO can be efficiently extended using the following strategies:\n* **Router conditioning:** The routing network can be conditioned on compressed representations of the weight vector (e.g., via projection or learned embeddings), avoiding a linear blowup in routing complexity.\n* **Expert reuse:** Experts can be shared across subsets of objectives, enabling parameter reuse across related regions of the Pareto front.\n* **Hierarchical routing:** For very high-dimensional cases, hierarchical or sparse gating strategies can further reduce expert activation and computational load.\n\n**W3: “Preference Learning” is vaguely defined.**\n\nThank you for the comment. We acknowledge that the definition of “Preference Learning” (PL) in the current version may not be sufficiently clear. In our context, PL refers to a learning paradigm where the model is trained to prefer one solution over another based on their scalarized objective values under a given weight vector. Specifically, instead of learning from scalar rewards (as in RL), the model receives pairwise supervision derived from the relative quality of two sampled solutions. This preference signal is then used to guide policy updates through a ranking-based loss.\n\nWe will revise the manuscript to include a more precise and formal definition of PL at its first mention and clarify its distinction from reinforcement learning approaches in both methodology and learning dynamics.\n\n\n**Q1: Does preference learning offer theoretical guarantees over REINFORCE?**\n\nThe majority of the work [1–2] within the neural MOCOP literature focused on the empirical side, making the provision of a solid theoretical analysis within the tight rebuttal period a non-trivial challenge. Here, we try to give an in-depth empirical analysis concerning the effects of preference learning (PL) method, with the aim of offering valuable insights for future research.\n\nCompared to standard REINFORCE-based reinforcement learning, our PL approach offers significant practical advantages. Instead of relying on high-variance reward signals (as in REINFORCE), PL compares solutions under the same weight vector, providing a low-variance signal. We provide concrete evidence of this in Table D, which reports the average gradient variance in the first five batches of training POCCO-W under both RL and PL frameworks. Across three MOTSP settings, PL consistently exhibits two to four orders of magnitude lower variance, leading to faster convergence and more stable optimization dynamics.\n\n**Table D: Gradient variance in REINFORCE vs. preference learning.** \n| Problem Size | Batch | RL Variance | PL Variance |\n|--------------|:-----:|:----------------:|:----------------:|\n| Bi-TSP20      |   1   |     0.054648     |     0.000314     |\n|              |   2   |     0.039951     |     0.000252     |\n|              |   3   |     0.019038     |     0.000191     |\n|              |   4   |     0.009093     |     0.000115     |\n|              |   5   |     0.010742     |     0.000104     |\n| Bi-TSP50      |   1   |     0.474784     |     0.000142     |\n|              |   2   |     0.220530     |     0.000077     |\n|              |   3   |     0.140275     |     0.000048     |\n|              |   4   |     0.092114     |     0.000040     |\n|              |   5   |     0.065286     |     0.000031     |\n| Bi-TSP100     |   1   |    10.124832     |     0.000059     |\n|              |   2   |     6.355499     |     0.000039     |\n|              |   3   |     3.461700     |     0.000032     |\n|              |   4   |     1.755804     |     0.000021     |\n|              |   5   |     1.111246     |     0.000014     |\n\n\n**Q2: Do results generalize to larger-scale instances? What limits scalability?**\n\nTo evaluate the out-of-distribution generalization ability of our method, we report additional results on larger Bi-TSP instances (150 and 200 nodes) as well as benchmark instances (KroAB150 and KroAB200) in Appendix H and I, respectively. Furthermore, to demonstrate scalability, we include new experimental results on even larger problems including Bi-TSP with 300/500/1000 nodes and MOKP with 300/500/1000 items. These results are summarized in Table E and Table F. Due to computational constraints, we generate 5 random instances for each setting. As shown, POCCO-W consistently outperforms the baseline WE-CA in most large-scale cases.\n\n**Table E: HV on large Bi-TSP instances.**\n| Problem Size | WE-CA HV | POCCO-W HV |\n|--------------|:--------:|:----------:|\n| Bi-TSP300     | 0.7441   | **0.7458**     |\n| Bi-TSP500     | 0.7476   | **0.7592**     |\n| Bi-TSP1000    | 0.7186   | **0.7400**     |\n\n**Table F: HV on large MOKP instances.**\n| Problem Size | WE-CA HV | POCCO-W HV |\n|--------------|:--------:|:----------:|\n| Bi-KP300      | **0.6000**   | 0.5947     |\n| Bi-KP500      | 0.4244   | **0.5658**     |\n| Bi-KP1000     | 0.2439   | **0.8408**     |\n\nWe acknowledge, however, that the performance of our method may degrade when applied to extremely large-scale problems (e.g., 10,000-node MOTSP). This limitation is primarily due to several factors:\n* **Decoder length and exposure bias:** In very long decoding sequences, small prediction errors accumulate, reducing solution quality.\n* **Router generalization gap:** The context distribution in extremely large problems deviates from training data, making expert routing less effective.\n* **Expert capacity saturation:** A fixed number of experts may be insufficient to model the growing diversity of subproblem structures.\n\nWe consider scaling to ultra-large MOCOPs a valuable direction for future work and plan to explore hierarchical decoding, adaptive expert allocation, and curriculum-based training to further extend POCCO’s applicability.\n\n**Q4: Does preference pair generation introduce bias? Is generalization tied to training distribution?**\n\nOur method generates preference pairs by sampling two candidate solutions under the same weight vector and comparing their scalarized objective values. This process is inherently tied to the decomposition-based formulation of multi-objective optimization, where each weight vector defines a specific subproblem. Thus, the preference signal is locally defined within each subproblem and does not rely on global dominance relationships.\n\nRegarding potential bias: while it is true that the quality of the preference supervision depends on the candidate solution pool (e.g., from early-stage policies), we mitigate this by continuously sampling pairs from the evolving policy during training. This dynamic sampling ensures that preference pairs reflect the model’s current solution space and are not biased toward a fixed subset of the Pareto front.\n\nAs for generalization, Figs. 3-4, Appendices H–I, and the large-scale experiments in Q2 demonstrate strong out-of-distribution generalization across problem sizes. Our model is trained on a range of problem sizes $n \\in \\[20, 21, \\dots, 100\\]$ (and $\\[50, \\dots, 200\\]$ for Bi-KP), following [1-2], which showed that training across sizes leads to better generalization than training on a fixed size. We believe this range-based training enables the model to learn distributional patterns of subproblem structures, rather than overfitting to specific instance sizes. We will include an analysis of the effect of training size distribution in the final version to clarify this further.\n\n[1] Conditional neural heuristic for multiobjective vehicle routing problems, TNNLS, 2024.\n\n[2] Rethinking neural multi-objective combinatorial optimization via neat weight embedding, in ICLR 2025."}, {"review_id": "7esITDdy9W", "reviewer": "Reviewer_FKaN", "summary": "This paper proposes a plug-and-play multi-objective reinforcement learning (RL) framework, POCCO. It consists of two major components.\n(1) Conditional Computation Block (CCO): It is a sparse router that employs a gated network, combined with a Top-k operator, to enable the model to decide which experts to activate. This can ensure the width and depth while remaining efficient. This aims to improve the exploration over pure REINFORCE.  \n(2) Preference-driven Learning: This method is inspired by the Bradley-Terry likelihood, where the weighted sum of rewards is replaced with the average log-likelihood gap between two randomly selected complete solutions. This aims to lower the gradient variance.\n\nThe experiments are done on bi/tri-objective TSP, CVRP, and Knapsack problems. The results look promising, and the backbone is improved when enhanced by their method.", "questions": "1. It would be great if you had more experimental results on some more complex problems. The backbones are already performing quite well for now, so the improvements seem minor.\n\n2. For comparison, you have shown that with PL, your convergence is significantly faster than with pure REINFORCE. Then why not set the solving time the same for both the backbone and backbone + your method? This seems to be a better comparison.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n(1) The method is a plug-and-play framework, and the two mechanisms are simple and elegant. So it would be easy to adapt to most multi-objective RL frameworks.\n\nWeaknesses:\n(1) The test problems are a bit \"easy\" since the backbone alone has already been very close to the optimal results. So the improvement seems minor. See more in the questions.\n(2) Some experimental settings are questionable. See more in the questions.", "was_revised": true, "final_justification": "All my questions are resolved and I will keep my score unchanged.", "rebuttal": "We sincerely appreciate the reviewer for providing positive and constructive comments. Here are our detailed responses to your comments, where W denotes Weakness (W1-W2) and Q denotes Question (Q1-Q2).\n\n**W1 & Q1: The test problems are a bit \"easy\" since the backbone alone has already been very close to the optimal results. So the improvement seems minor. It would be great if you had more experimental results on some more complex problems.**\n\nWe acknowledge that the benchmark problems used in our main experiments—Bi-TSP, Tri-TSP, Bi-CVRP, and Bi-KP—are widely adopted in prior neural MOCOP work [1–5], which facilitates fair comparison with existing baselines. However, we agree that these instances may not fully reflect the challenges posed by real-world constraints.\n\nFollowing the reviewer’s suggestion to address the applicability of our method under more complex constraints, we conducted an additional experiment on the multi-objective capacitated vehicle routing problem with time windows (MOCVRPTW), as summarized in Table G. In the MOCVRPTW, each customer node $v_i$ is associated with a time window $[e_i, l_i]$ and a service time $s_i$[6]. A vehicle must start serving customer $v_i$ in the time slot from $e_i$ to $l_i$.  If the vehicle arrives earlier than $e_i$, it has to wait until $e_i$. All vehicles must return to the depot $v_0$ no later than $l_0$. This study considers three objectives: minimizing the total travel distance, minimizing the number of routes, and minimizing the average route length per customer served. As shown in Table G, POCCO-W consistently outperforms WE-CA on MOCVRPTW tasks as well.\n\n**Table G: HV on MOCVRPTW Instances.**\n| Problem Size | WE-CA HV | POCCO-W HV |\n|--------------|:--------:|:----------:|\n| MOCVRPTW20    | 0.5465   | **0.5496** |\n| MOCVRPTW50    | 0.5499   | **0.5576** |\n| MOCVRPTW100   | 0.5331   | **0.5375** |\n\n**W2 & Q2: Why not set the solving time the same for both the backbone and backbone + your method? This seems to be a better comparison.**\n\nThanks for your suggestion. We have added a new experiment to compare POCCO-W with the backbone method (WE-CA) under the same solving time constraint. Specifically, we limit each instance to a 10-second solution sampling budget for both methods and evaluate on the same test datasets used in the main paper (200 instances per problem setting). As shown in Table M, POCCO-W consistently outperforms WE-CA across all benchmark problems, even when both are given exactly the same solving time. We will include a comprehensive comparison under the same solving time in the final version of the paper.\n\n**Table M: HV of POCCO-W and WE-CA Under Equal Solving Time (10s per instance).**\n| Problem     | WE-CA  | POCCO-W  |\n|-------------|:--------------------:|:----------------------:|\n| Bi-TSP100   | 0.7049               | **0.7071**                 |\n| MOCVRP100   | 0.4076               |  **0.4090**               |\n| Bi-KP100    | 0.4531               | **0.4532**                |\n| Tri-TSP100  | 0.4999               |  **0.5022**                |\n\n[1] Pareto set learning for neural multi-objective combinatorial optimization, in ICLR 2022.\n\n[2] Neural multi-objective combinatorial optimization with diversity enhancement, in NeurIPS 2023.\n\n[3] Conditional neural heuristic for multiobjective vehicle routing problems, TNNLS, 2024.\n\n[4] Collaborative deep reinforcement learning for solving multi-objective vehicle routing problems, in AAMAS 2024.\n\n[5] Rethinking neural multi-objective combinatorial optimization via neat weight embedding, in ICLR 2025.\n\n[6] MVMoE: Multi‑Task Vehicle Routing Solver with Mixture‑of‑Experts, in ICML 2024."}]

Author rebuttal:

Meta-review: This study introduces a new architecture for multi-objective combinatorial optimization using deep reinforcement learning, featuring two key components: conditional computation and preference learning. The main contribution of this research is empirical evidence that the POCCO architecture outperforms all baseline models across several benchmark tests.

Overall, the paper has received mixed reviews, ranging from borderline acceptance to clear endorsement. Furthermore, the authors addressed both theoretical and experimental concerns effectively during the rebuttal phase. Therefore, I recommend acceptance of the paper.

In the revised version, it would be beneficial to include the improvements made during the rebuttal phase. Specifically, the related work section should be integrated into the main paper. Additionally, further experiments covering important aspects—such as additional baselines, the impact of the number of experts, top-k selection of experts, and real-world constraints—could be included either in the main paper or the appendix. Finally, the theoretical analysis should also be provided in the appendix.
Decision: Accept (poster)