Title: Fair Minimum Labeling: Efficient Temporal Network Activations for Reachability and Equity
Venue: NeurIPS 2025
Accepted: yes
Keywords: fairness

temporal graphs

reachability

resource efficiency

approximation algorithms

optimization

group-level coverage

multi-source learning

communication-constrained learning

Green AI
Abstract: Balancing resource efficiency and fairness is critical in networked systems that support modern learning applications. We introduce the _Fair Minimum Labeling_ (FML) problem: the task of designing a minimum-cost temporal edge activation plan that ensures each group of nodes in a network has sufficient access to a designated target set, according to specified coverage requirements. FML captures key trade-offs in systems where edge activations incur resource costs and equitable access is essential, such as distributed data collection, update dissemination in edge-cloud systems, and fair service restoration in critical infrastructure. We show that FML is NP-hard and $\Omega(\log |V|)$-hard to approximate, where $V$ is the set of nodes, and we present probabilistic approximation algorithms that match this bound, achieving the best possible guarantee for the activation cost. We demonstrate the practical utility of FML in a fair multi-source data aggregation task for training a shared model. Empirical results show that FML enforces group-level fairness with substantially lower activation cost than baseline heuristics, underscoring its potential for building resource-efficient, equitable temporal reachability in learning-integrated networks.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Fair Minimum Labeling (FML) problem, which aims to compute temporally sparse edge activation schedules in dynamic networks while ensuring fair group-wise reachability to a central terminal under global resource constraints. The FML framework formalizes the trade-off between minimizing the number of activated edges and satisfying fairness constraints across multiple groups, specifically requiring that a fixed fraction of nodes in each group temporally reach the terminal. \n\nThe authors show that the problem is NP-hard and logarithmically hard to approximate. To address this, they propose a randomized approximation framework based on probabilistic tree embeddings, leading to two algorithms: (1) an exact method on trees and (2) a more scalable bicriterial approximation that allows bounded fairness violations. \n\nBoth methods are evaluated in a multi-source learning task under resource constraints, demonstrating that FML-based solutions outperform fairness-agnostic baselines in terms of cost-efficiency and fairness. The paper also discusses practical limitations and outlines future directions, including extensions to multi-terminal and distributed settings.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n* The paper introduces the Fair Minimum Labeling (FML) problem, which uniquely captures temporal fairness constraints in dynamic networks—a formulation not addressed in prior work.\n* The authors provide formal NP-hardness and approximation lower bounds for the problem, and derive approximation algorithms, indicating tight theoretical contributions.\n* The use of probabilistic tree embeddings, combined with both exact and bicriteria approximation algorithms, demonstrates a thoughtful adaptation of known techniques to a new fairness-driven setting.\n* The experiments are clearly motivated and show that FML-based solutions outperform baselines in both activation cost and fairness, with controlled runtime tradeoffs.\n\nWeakness:\n* While the FML framework is motivated by real-world applications such as federated data collection, edge-cloud communication, and fair infrastructure restoration, the current formulation only supports two demographic groups and a single terminal. This limitation restricts the applicability of the proposed methods to more complex, realistic settings that involve multiple groups or multiple target nodes.\n* The FML framework assumes centralized control and fixed fairness thresholds, which may not directly translate to more dynamic, decentralized environments commonly seen in federated or edge settings.\n* The exact dynamic programming solution has a runtime of $\\mathcal{O}(n^5)$, making it impractical for large-scale networks without further optimization. The bicriteria algorithm sacrifices exact fairness, and its violation factor grows logarithmically with the graph size, which may be undesirable in sensitive applications. \n* The framework focuses solely on group-level static fairness (i.e., coverage quotas), without addressing finer-grained notions such as fairness over time or at the individual level.\n* While synthetic experiments are reasonable, the evaluation would be stronger if applied to a real-world temporal network with fairness-sensitive requirements.\n\nQUESTIONS:\n* The current cost model treats all edge activations equally by summing $|\\lambda(e)|$ across edges. However, in real-world scenarios, activation costs may vary—e.g., long-distance links may consume more energy, low-bandwidth edges may incur higher latency, or peak-time activations may be costlier. Can the FML framework be extended to support weighted activation costs, and what challenges might arise in doing so?\n* The current model enforces strictly increasing timestamps along temporal paths. In practice, however, multiple edges may be traversed simultaneously, or edge activations may involve delays (e.g., arriving at time $\\tau + d$). Have you considered relaxing this assumption to allow non-strict time progression or modeling delays explicitly? \n* The approximation factor of $O(\\log n)$ is shown to be tight in expectation via FRT embeddings. Can you clarify whether this tightness carries over in worst-case performance as well?\n* The framework uses a shared $\\alpha$ for all groups. Can different coverage ratios (e.g., $\\alpha_B$, $\\alpha_R$) be supported? Would this affect the approximation guarantees or algorithm design?\n* The fairness guarantee depends on tree height $H$. Do you have empirical data (e.g., typical values of $H$ or fairness violation) showing how this impacts coverage in practice?\n* The terminal node $t$ is fixed. Since terminal placement affects reachability and cost, have you studied how the choice of $t$ (e.g., central vs. peripheral) influences fairness and performance?\n* The exact DP algorithm has $O(n^5)$ time complexity, and the bicriteria method improves this to $O(n^2 + n\\varepsilon^{-4} \\log^4 n)$. Do you have empirical benchmarks on runtime and scalability, especially for larger graphs (e.g., $n > 10^4$)? Could sparsity or graph structure be exploited to further improve scalability?\n* The current model supports only two groups and one terminal. Can it be extended to multiple groups or terminals? What new challenges would this introduce, and how would it affect scalability?\n* The model assumes a static node set. In dynamic environments (e.g., nodes dropping out), how robust is FML? Can it be adapted to handle such changes without full recomputation?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 4, "DATE": "2025-06-28"}

comments: SUMMARY:
This paper introduces the Fair Minimum Labeling (FML) problem, addressing group-wise temporal reachability under resource constraints. It proposes a probabilistic approximation framework using tree embeddings, ensuring efficient cost and fairness for two distinct groups reaching a single terminal.

STRENGTHS AND WEAKNESSES:
# Strengths
- The paper addresses a task with clear real-world motivation and potential application value.
- The theoretical formulation of FML and its solution is detailed and well-articulated, demonstrating a strong analytical foundation, which contribute meaningfully to the understanding of the proposed method.
# Weaknesses
- The current FML assumptions and the experiment setup appear somewhat idealized, raising questions about the general applicability and robustness of the method in practical settings.
- The proposed approach introduces a notable increase in computational time, as shown in Table 1, which could become a limiting factor for real-time or resource-constrained applications. This trade-off between performance and efficiency should be more clearly discussed.
- There seems to be a discrepancy between the results reported in Table 1 (specifically for the "approx" and "biapprox" variants) and the corresponding description in Line 344 of the manuscript.

QUESTIONS:
- Have the authors explored more potential strategies or optimizations that could reduce the computational overhead?
- Include a more dedicated discussion on the limitations of the current assumptions and explore how they might be relaxed or adapted to more realistic environments like the mentioned distributed and federated learning.
- Double-check and ensure consistency across all quantitative results and textual interpretations.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a new framework, termed Fair Minimum Labeling, for finding minimum-cost (temporal) paths between sources and target nodes in a graph while ensuring that the number of source nodes from different groups (types) is fair (are similar). The paper shows that the underlying optimization is NP-hard and proposes two approximate algorithms to solve the problem above. The numerical results show the method's effectiveness in lower cost and fair coverage.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n- The paper is well-written, and the problem statement is clear. The problem of network optimization with fairness constraints is timely and well-motivated from a theoretical and practical perspective.\n- The proofs and the math look ok. The theoretical results are precise and essential to motivate the design of approximated algorithms. \n- The solution based on probabilistic tree embeddings is principled, which leverages tools from metric space for solving optimization problems on graphs.\n- The authors provide code to reproduce the experiments.\n\nWeaknesses: \n- Lack of numerical experiments on real data to support the results and additional experiments to showcase the method's utility.\n- Complexity of order $O(n^5)$.  Although the bicriteria approximation reduces the complexity, this comes at the cost of a dependency on the guarantees of the tree height\n- The proposed solutions are limited to a single terminal and two distinct groups, making the whole introduction too complex for the final method.\n- The paper assumes a central controller can decide when communication links are active. This significant assumption may not hold in decentralized or uncoordinated settings, such as classical federated learning, where device participation is often uncoordinated. \n- A few citations on fairness on graphs are missing [1], [2]\n\n[1] Navarro, Madeline, et al. \"Fair GLASSO: Estimating Fair Graphical Models with Unbiased Statistical Behavior.\" Advances in Neural Information Processing Systems, 2024\n[2] Zhou, Zhuoping, et al. \"Fairness-aware estimation of graphical models.\" Advances in Neural Information Processing Systems, 2024\n\nQUESTIONS:\nI would like to see an experiment on a real-world case, not just with simulation data. I think verifying the method in a more realistic dataset will strengthen the submission. I am open to increasing the score with stronger numerical results, with a real dataset (not necessarily something demanding considering the short time of the rebuttal)\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper introduces the Fair Minimum Labeling (FML) problem, motivated by applications like fair data collection or model updates in edge-cloud systems. The authors prove FML is NP-hard, then propose probabilistic approximation algorithms achieving tight approximation bound. Empirical evaluation on multi-source learning tasks shows that FML-based methods enforce group fairness with significantly lower activation cost than baseline heuristics while maintaining high accuracy across groups.

STRENGTHS AND WEAKNESSES:
**Strength**

+) This paper proposes an interesting and practical problem which can be used in multiple real scenrios

+) This paper proved NP-hardness and found an algorithm to reach the theoretical bound

+) The effectiveness of the method is verified on practical problems


**Weakness**

-) Experiments are limited to synthetic random geometric graphs with two synthetic data groups. It would be better to evalute on real-world network topologies.

-) Even the approximate algorithm has runtimes quadratically which may prevent it usage in practice. It would be better to discuss if more approximation/assumptions can be applied to make it more practical?

QUESTIONS:
a) Can this work be extended with more approximation/assumptions can be applied to make it more practical in runtime?

b) Can this work be used to real-world scale of network topology?

Note: I am not an expert in this area and my review focuses on the practical impact of this theoretical work. I am neutral accepting the paper, and would love more discussions from other reviewers.

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new problem: Fair Minimum Labeling (FML) which lies in the intersection of temporal graph algorithms and fairness in decision making. The problem asks for the minimum cost for labeling edges, each at a set of timestamps, on a given graph, such that at least a fixed portion of nodes in each color group can \"temporally reach\" (meaning there exists a path of edges with ) a given set of terminal nodes. The paper then gives a concrete algorithm (and its more scalable approximation version) to solving such a problem in the case of two color groups and singleton terminal node. It completes the story with proving the hardness of the problem, even for singleton node case with only one terminal. The results are complemented with experiments with practical tasks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The problem is new and seems to have practical application, so I think it is definitely worth studying. Especially, introducing fairness into problem formulation should be important for such a problem.\n- The study is quite complete (at least with the given assumptions). The authors have proposed the problem, proved the hardness the found solutions that match the performance bounds. \n- The algorithm seems easy to implement, so potentially usable.\n\nWeaknesses:\n- The experiments are somewhat less convincing than the theory. My major concern is that it seems too slow, especially compared to the benchmarks. Further, the self-synthesized datasets seem a bit too extreme, as it is generating two clusters of points that are in general very far away from each other. As a result GREEDY completely overlooks one of them. \n- The baselines are limited, partially attributed to this problem being newly proposed. However, the chosen baselines clearly have some deficiency in either the fairness or cost perspective. GREEDY is bad for fairness because it doesn't account for it. CLOSEST is bad for cost because it blindly activates the edges along the shortest path (so it can use the same edge multiple times). Is there no simple baseline that seems to make sense for both?\n- The algorithm uses some nice techniques. I'm not an expert in temporal graphs so I'm not sure how novel they are given the temporal graph contexts, but it seems to lack technical depth.\n\nQUESTIONS:\n1. I don't have much temporal graph context. I can kind of see its practical use, but it would still be nice to have a concrete example where this kind of graph is used, and show what the nodes and the temporal activation of edges embody for a real-life problem, and why the temporal information matters.\n2. Are the results in Sec. 5 limited to the two-colors-plus-no-color case only? It seems the derivation of run-time uses that.\n3. Do you know how the run-time scales with size of graph on the real datasets? While theory suggests polynomial run-time, in practice it could be different. How did you choose the graph size in the experiments?\n4. Do you have the de-composition of your algorithms? They seem to be taking way more time than the baselines with improvements in performance. Do we know if the tree-embedding or the labeling algorithm takes more time?\n5. If I understand things correctly, in the case of singleton terminal, essentially we are solving a fair connectivity problem on the static graph. While the reduction is simple and effective, the algorithm itself is not concerned with any temporal information. Is that correct? Hence we kind of lose the temporal aspect of the problem. Have you any thoughts on how to solve this problem in the multi-node terminal case?\n6. If the terminal is singleton, is this problem essentially equivalent to finding a minimum cost tree that spans at least a given portion of nodes of both colors? The resulting chosen edges seem to always form a tree that connects a chosen set of nodes to the terminal node.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-05"}