Title: Learning-Augmented Online Bipartite Fractional Matching
Venue: NeurIPS 2025
Accepted: yes
Keywords: Learning-augmented algorithms

algorithms with predictions

algorithms with advice

robustness-consistency tradeoff

online bipartite matching

AdWords

online algorithms
Abstract: Online bipartite matching is a fundamental problem in online optimization, extensively studied both in its integral and fractional forms due to its theoretical significance and practical applications, such as online advertising and resource allocation. Motivated by recent progress in learning-augmented algorithms, we study online bipartite fractional matching when the algorithm is given advice in the form of a suggested matching in each iteration. We develop algorithms for both the vertex-weighted and unweighted variants that provably dominate the naive ``coin flip'' strategy of randomly choosing between the advice-following and advice-free algorithms. Moreover, our algorithm for the vertex-weighted setting extends to the AdWords problem under the small bids assumption, yielding a significant improvement over the seminal work of Mahdian, Nazerzadeh, and Saberi (EC 2007, TALG 2012). Complementing our positive results, we establish a hardness bound on the robustness-consistency tradeoff that is attainable by any algorithm. We empirically validate our algorithms through experiments on synthetic and real-world data.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies online bipartite fractional matching with advice and proposes two learning-augmented algorithms—LAB for vertex-weighted and PAW for unweighted settings—that provably outperform existing methods. Both algorithms dominate the naive COINFLIP baseline across the entire robustness-consistency tradeoff, a first in this line of work. LAB further extends to the AdWords problem under the small bids assumption, improving upon classic results by Mahdian et al. The paper also establishes new theoretical upper bounds by constructing hard instances that reveal fundamental limits of any algorithm in this framework. Extensive experiments on synthetic and real-world graphs confirm the algorithms’ effectiveness: they achieve optimal performance under accurate advice and gracefully degrade as advice quality worsens, aligning with theoretical predictions. This work advances the theory and practice of learning-augmented online optimization.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths and Weaknesses:\n\nThis paper presents a thorough and impactful contribution to the field of online optimization. It introduces two learning-augmented algorithms, LAB and PAW, for online bipartite fractional matching in both vertex-weighted and unweighted settings. A key strength is the theoretical breakthrough: both algorithms provably dominate the COINFLIP baseline across the entire robustness-consistency tradeoff spectrum—a significant advancement over prior works, which only achieved partial domination. Additionally, the extension of LAB to the AdWords problem under the small bids assumption enhances its practical applicability, improving upon the classic Mahdian-Nazerzadeh-Saberi framework. The authors further provide tight upper bounds on what any algorithm can achieve via a novel hardness construction, and back up their theoretical findings with comprehensive experiments on both synthetic and real-world datasets, demonstrating smooth performance degradation with increasing noise in the advice.\n\nHowever, the paper’s technical exposition could be improved. While the core algorithmic ideas and analysis are sound, several important definitions and procedural steps are presented in a terse or abstract way, making the work less accessible to readers unfamiliar with the topic. For example, the intuition behind the penalty functions and dual constructions could be elaborated with more concrete examples or illustrations. Strengthening the clarity and pedagogical structure of the technical sections—particularly the algorithm descriptions and analytical proofs—would make the paper more readable and broaden its impact beyond the expert audience.\n\nQUESTIONS:\nNo.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper under consideration studies the vertex-weighted online bipartite fractional matching problem with predictions. That is, there is a bipartite graph where one side is available to the algorithm upfront offline. Every offline vertex is assigned a weight (if all weights coincide, the instance reduces to an unweighted one). The other side of vertices arrives online one-by-one. \nIn the purely adversarial setting, nodes arrive without any information and the decision maker needs to immediately and irrevocably decide to which extend an edge is (fractionally) added to the matching. \nIn the setting with predictions, there is a prediction for each online vertex indicating to which extend each incident edge should be considered.\n\nThe paper derives algorithms for the unweighted and vertex-weighted setting and derives results for robustness and consistency---a classical approach in the algorithms with predictions literature. More precisely, for the vertex-weighted variant, a learning-augmented version of the balance algorithm is used to derive the robustness-consistency guarantees. For the unweighted case, an extension of the classical waterfilling algorithm is developed. \n\nFrom a technical perspective, the analysis is based on a primal-dual approach. In particular, depending on the arrivals, primal and dual variables are updated in a suitable way to maintain feasible solutions for both problems. These can then be used to apply typical primal-dual arguments for competitive ratios. \n\nThe authors can extend their results also to the AdWors problem when imposing a small-bid assumption and complement results with an upper bound for the integral-advice case.\n\nSTRENGTHS AND WEAKNESSES:\n*Strengths*:\nThe paper derives reasonable guarantees for robustness and consistency in a variety of settings in online bipartite fractional matching. In particular, the guarantees outperform trivial bounds from coinflip arguments and are extendable to more general settings (as AdWords, with a few reasonable assumptions)---indicating the power of this approach. The primal-dual analysis is nice and clean, making it an enjoyable read.\n\n*Weaknesses*: \nThat said, the paper is required to use different algorithms for the different problem formulations---tailoring the approach to the respective problem. The design of the algorithms feels natural, but not really surprising. Given this design, the primal-dual approach for the analysis is expectable. \nHaving a unified approach for both cases (unweighted and vertex-weighted) would be a very desirable goal.\n\nQUESTIONS:\n* One of my main concerns is the novelty and originality of the paper: Even though I like the general approach, I got the impression that this work only adds marginal value to the huge body of literature in the algorithms-with-predictions regime. Can you clearly state the main differences why this work is adding sufficiently compared to previous work, please? (In the paper, there is only a small paragraph on page 2 on this unfortunately)\n\n* In the paper, you are required to use two algorithms for the vertex-weighted and unweighted case. Do you see a chance of unifying both settings? More precisely, what is the deeper intuition why these two approaches are required?\n\n* As a follow-up point, can it be the case that the primal-dual analysis is simply too lossy to get reasonable guarantees in both cases?\n\nLIMITATIONS:\nNo negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles the problem of online bipartite fractional matching with the support of advice. In this setting, there exists a bipartite graph between a set of offline vertices $U$ that have a value associated with them and a set of online vertices $V$ that arrive one by one. Upon the arrival of each online vertex, its neighbors along with suggested fractional matchings to its neighbors are revealed. The algorithm must then decide, in real time and without future knowledge, how to match (fractionally) the arrived online vertex to its neighbors subject to feasibility constraints.\n\nThe authors present two novel algorithms:\n\n* LEARNINGAUGMENTEDBALANCE (LAB) for the vertex-weighted setting, which adapts the classic BALANCE algorithm using an advice-aware penalty function.\n* PUSHANDWATERFILL (PAW) for the unweighted setting, which introduces a two-phase approach combining targeted flow pushing and traditional waterfilling.\n\nA key contribution is that both algorithms strictly improve upon the baseline COINFLIP algorithm across the entire robustness-consistency tradeoff spectrum, which quantifies the balance between worst-case guarantees and performance under accurate advice.\n\nThe LAB algorithm is also shown to improve upon prior work for the AdWords problem under the small bids assumption, offering better guarantees than the well-known Mahdian et al. framework. On the other hand, PAW is tailored to overcome analysis challenges specific to the unweighted setting and shows strong performance both theoretically and empirically.\n\nAdditionally, the paper provides a new upper bound on the robustness-consistency tradeoff, which is shown to be tight in many cases and improves over previous results. Experimental evaluations on both synthetic and real-world bipartite graphs demonstrate that the proposed algorithms interpolate smoothly between advice-based and advice-free strategies, with performance aligning with theoretical predictions.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n* The paper presents two new learning-augmented algorithms: LAB, which modifies the BALANCE algorithm with advice-aware penalties, and PAW, which adds a controlled \"push\" step to WATERFILLING for unweighted graphs, achieving better tradeoffs than COINFLIP.\n* The paper develops a factor-revealing LP to derive hardness results, improving prior bounds.\n* Experiments across a variety of datasets and noise levels confirm theoretical insights.\n\n**Weaknesses**\n* There is an assumption that the advice to the PAW algorithm is integral, while the algorithm generates fractional matchings. \n* I have some concerns with the experimental setup. In particular, I like the idea of having a parameter $\\gamma$ controlling the accuracy of the provided advice; however, the same level of noise is not reflected in the CR of the algorithms simply because the advice matches the online vertices to non-neighbor offline vertices, which are infeasible and the LAB algorithm corrects the infeasibility by matching to other points. (To better understand my concern, see Figure 4: when $\\lambda=1$ (the LAB fully trusts the advice and completely follows it) and $\\gamma=1$ (fully noisy setting where advice is completely irrelevant), the CR is still above 0.9 in three cases). I believe the experiments are more meaningful if the algorithm is tested on valid but poor advice. For instance, I expect to see a CR of 0.63 or so when $\\lambda=1$ and the advice provides a 1/2-approx matching.\n\nQUESTIONS:\n* In each experiment, it seems like there exists a special value $\\gamma^*$ such that at that level of noise, the CR of all executions of LAB and PAW (with different values of $\\lambda$) and the BALANCE algorithm coincide. This can be seen in the experiments in the appendix as well. Could the authors clarify if there is a theory behind this observation and whether this convergence has theoretical significance? Is this due to advice being entirely uncorrelated at that γ, leading LAB to behave equivalently to BALANCE regardless of λ?\n\n* Can the authors clarify the reason why the PAW algorithm does not extend to the case where the advice is also fractional?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

comments: SUMMARY:
In this paper, the authors consider online bipartite fractional matching problems when the online decision maker is provided advice in the form of a suggested (offline feasible) matching in each iteration. They design algorithms for both the vertex-weighted and unweighted versions of the problem that are provably better than the coin-flipping strategy of randomly choosing between advice-following and advice-free algorithms. They also show that the vertex-weighted version can be extended to the AdWords problem with small bids. They also provide hardness bound on the robustness-consistency tradeoff and validate their algorithms through synthetic and real-world data.

STRENGTHS AND WEAKNESSES:
Strengths:
- Results, claims and proofs are technical sound and solid and provide a good picture on what can be achieved for these problems under learning-augmented algorithms.
- Paper is very well written, without going into unnecessary technical details within the main body.
- Content of the paper provides additional and novel aspects to the literature on online bipartite fractional matching problems under learning-augmented algorithms.
- Novel and well positioned paper.

Weaknesses:
- Would have been nice to get additional insights on the open problems listed in the conclusion, but this is a minor weakness, as I liked the paper ...

QUESTIONS:
The authors list several interesting open questions in their conclusion. I wonder if they have any comments indicating how some of the content and technical ideas within the paper could be extended or be helpful (or not) in addressing these questions.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-30

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies learning-augmented algorithms for online bipartite *fractional* matching, aiming to outperform the COINFLIP baseline by achieving better robustness–consistency tradeoffs.\n\nIn online bipartite matching, a bipartite graph is revealed incrementally: the offline vertices are known in advance, while online vertices arrive one by one, requiring irrevocable matching decisions to maximize the total matched weight. (In the vertex-weighted case, this means maximizing the sum of the offline vertex weights.)\n\nThe work focuses on a predictive setting where, upon each arrival, the algorithm receives advice suggesting a fractional matching. The algorithm seeks to remain robust to poor advice while exploiting accurate predictions.\n\nThe authors propose two algorithms: LAB for the weighted case and PAW for the unweighted case, and establish hardness results that delineate the limits of such tradeoffs.\n\n\nThe COINFLIP algorithm randomly chooses between an advice-free strategy that guarantees worst-case performance and an advice-following strategy that assumes the predictions are perfectly accurate, effectively tracing a linear tradeoff curve between robustness and consistency.\nThey show that there exists a learning-augmented algorithm for online bipartite matching that dominates COINFLIP across the entire range of robustness, providing improved guarantees regardless of how accurate the advice is.\n\nThey mainly use a primal-dual analysis framework to design and analyze their algorithms, adapting classical techniques to incorporate advice-dependent penalty functions that balance following predictions and maintaining worst-case guarantees.\n\n\nFor the vertex-weighted case, they use a version of a continuous process that incrementally pushes flow to neighbors based on an advice-aware carefully chosen penalty function, prioritizing offline vertices with higher weights and those recommended by the advice.\n\nFor the unweighted case, they propose a different algorithm called PUSHANDWATERFILL, which first pushes flow to the advised offline vertex up to a specified threshold and then distributes any remaining flow using the standard waterfilling strategy, allowing better robustness-consistency tradeoffs than COINFLIP in this setting.\n\nThey also improve the upper bound by designing a hardness construction using two adaptive adversaries, one targeting robustness and the other consistency, which remain indistinguishable during the initial phase, making it provably impossible for any algorithm to exceed certain tradeoffs across the entire spectrum.\n\n\n\n\nThe COINFLIP algorithm randomly chooses between an advice-free strategy that guarantees worst-case performance and an advice-following strategy that assumes the predictions are perfectly accurate, effectively tracing a linear tradeoff curve between robustness and consistency.\nThey show that there exists a learning-augmented algorithm for online bipartite matching that dominates COINFLIP across the entire range of robustness, providing improved guarantees regardless of how accurate the advice is.\n\nThey mainly use a primal-dual analysis framework to design and analyze their algorithms, adapting classical techniques to incorporate advice-dependent penalty functions that balance following predictions and maintaining worst-case guarantees.\n\n\nFor the vertex-weighted case, they use a version of a continuous process that incrementally pushes flow to neighbors based on an advice-aware carefully chosen penalty function, prioritizing offline vertices with higher weights and those recommended by the advice.\n\nFor the unweighted case, they propose a different algorithm called PUSHANDWATERFILL, which first pushes flow to the advised offline vertex up to a specified threshold and then distributes any remaining flow using the standard waterfilling strategy, allowing better robustness-consistency tradeoffs than COINFLIP in this setting.\n\nThey also improve the upper bound by designing a hardness construction using two adaptive adversaries, one targeting robustness and the other consistency, which remain indistinguishable during the initial phase, making it provably impossible for any algorithm to exceed certain tradeoffs across the entire spectrum.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper makes a strong contribution to learning-augmented online optimization by introducing algorithms that achieve improved robustness-consistency tradeoffs across the entire spectrum. The authors address one of the most fundamental problems in online algorithms—online bipartite matching—combining elegant algorithm design with rigorous primal-dual analysis and an improved hardness construction that clarifies the theoretical limits of the problem. Empirical results on diverse datasets convincingly support the theoretical claims.\n\nI recommend accepting this paper as it advances both the theoretical foundations and practical understanding of learning-augmented online matching.\n\nQUESTIONS:\nNone.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}