OpenNovelty.org

Efficient algorithms for Incremental Metric Bipartite Matching

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 7.0Download Report PDF
metric bipartite matchingdynamic algorithm1-Wasserstein distance
Abstract:

The minimum-cost bipartite matching between two sets of points RR and SS in a metric space has a wide range of applications in machine learning, computer vision, and logistics. For instance, it can be used to estimate the 11-Wasserstein distance between continuous probability distributions and for efficiently matching requests to servers while minimizing cost. However, the computational cost of determining the minimum-cost matching for general metrics spaces, poses a significant challenge, particularly in dynamic settings where points arrive over time and each update requires re-executing the algorithm. In this paper, given a fixed set SS, we describe a deterministic algorithm that maintains, after ii additions to RR, an O(1/δ0.631)O(1/\delta^{0.631})-approximate minimum-cost matching of cardinality ii between sets RR and SS in any metric space, with an amortized insertion time of O~(n1+δ)\widetilde{O}(n^{1+\delta}) for adding points in RR. To the best of our knowledge, this is the first algorithm for incremental minimum-cost matching that applies to arbitrary metric spaces.

Interestingly, an important subroutine of our algorithm lends itself to efficient parallelization. We provide both a CPU implementation and a GPU implementation that leverages parallelism. Extensive experiments on both synthetic and real world datasets showcase that our algorithm either matches or outperforms all benchmarks in terms of speed while significantly improving upon the accuracy.

Disclaimer
This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper presents a deterministic algorithm for maintaining an approximate minimum-cost bipartite matching in arbitrary metric spaces as points arrive incrementally. It occupies the 'Incremental and Dynamic Matching Algorithms' leaf of the taxonomy, which contains only this paper as a sibling. This isolation suggests the leaf represents a relatively sparse research direction within the broader field of online and dynamic matching, distinguishing itself from the more populated branches focused on immediate irrevocable decisions or delay-based models.

The taxonomy reveals that most prior work concentrates on online matching with immediate decisions (eight papers across deterministic and randomized approaches in general metrics) or matching with delays (eleven papers across general metrics, bipartite constraints, and non-linear costs). The original paper's position in a separate incremental branch indicates it diverges from these paradigms by allowing reoptimization as the input evolves, rather than committing irrevocably or trading off delay costs. Neighboring leaves like 'Online Matching with Recourse' (two papers) share some conceptual overlap in permitting post-hoc adjustments, but the incremental setting appears to emphasize gradual input growth over bounded recourse budgets.

Among the three contributions analyzed, the literature search examined fourteen candidate papers total. The first contribution (constant-factor approximation with sublinear update time) examined two candidates with zero refutations. The second contribution (push-relabel framework for hierarchical metrics) also examined two candidates with zero refutations. The third contribution (parallelizable design with CPU/GPU implementations) examined ten candidates, again with zero refutations. These statistics suggest that within the limited search scope—focused on top semantic matches and citations—no prior work was identified that directly overlaps with the specific combination of incremental setting, general metric spaces, and the proposed algorithmic techniques.

Based on the top-fourteen semantic matches examined, the work appears to occupy a distinct niche: incremental maintenance of approximate matchings in arbitrary metrics with efficient updates. The absence of sibling papers in the same taxonomy leaf and zero refutations across all contributions indicate novelty relative to the surveyed literature. However, the limited search scope means the analysis does not cover the full landscape of dynamic graph algorithms or specialized metric structures that might contain related techniques outside the immediate matching literature.

Taxonomy

Core-task Taxonomy Papers
26
3
Claimed Contributions
14
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: incremental minimum-cost bipartite matching in metric spaces. The field addresses how to match arriving requests to servers or resources in a metric space while minimizing total cost, under varying constraints on when and how matching decisions can be made or revised. The taxonomy organizes work into several main branches: Online Matching with Immediate Irrevocable Decisions focuses on classical models where each request must be matched instantly and permanently, often yielding competitive ratio bounds that depend on metric properties (e.g., Randomized Minimum Metric[19], Log2k Competitive Metric[21]). Online Matching with Delays permits deferring decisions for a bounded time, trading off waiting costs against better information (e.g., Bipartite Matching Delays[9], Polylogarithmic Delays[22]). Online Matching with Recourse allows limited changes to prior assignments, exploring how much improvement recourse can provide (e.g., Min Cost Recourse[7], Line with Recourse[2]). Incremental and Dynamic Matching Algorithms study settings where the input evolves more gradually or where offline-style reoptimization is feasible (e.g., Optimal Weighted Bipartite[6]). Finally, Algorithmic Foundations and Related Problems encompass broader theoretical tools, smoothed analysis, and connections to problems like k-server (e.g., Empirical k-Server[26], Smoothed Single Sample[10]). A particularly active theme contrasts the power of delays versus recourse: works like Metric Matching Delay[13] and Deterministic Delays[11] show that even modest waiting can significantly improve competitive ratios, while studies such as Online Line Recourse[12] and Haste Makes Waste[8] explore how allowing a few reassignments can yield similar gains. Another line examines input-sensitive or beyond-worst-case guarantees (Beyond Worst Case[1], Input Sensitive Analysis[25]), moving away from purely adversarial models. The original paper, Incremental Metric Bipartite[0], sits within the Incremental and Dynamic Matching Algorithms branch, suggesting a focus on settings where the input arrives more predictably or where reoptimization is permitted. Compared to immediate-decision models like Randomized Minimum Metric[19] or delay-based approaches like Bipartite Matching Delays[9], Incremental Metric Bipartite[0] likely emphasizes maintaining or updating matchings as the instance grows, rather than committing irrevocably or waiting within strict time windows.

Claimed Contributions

First constant-factor approximation algorithm for incremental metric bipartite matching with sublinear update time

The authors introduce a deterministic algorithm that maintains an O(1/δ^0.631)-approximate minimum-cost matching in arbitrary metric spaces with amortized insertion time of O(n^(1+δ)). This is the first algorithm for incremental minimum-cost matching that applies to arbitrary metric spaces while guaranteeing provably fast updates.

2 retrieved papers
Push-relabel framework for maintaining 1-feasible matchings across hierarchical metric levels

The authors develop a novel push-relabel framework that maintains partial 1-feasible matchings simultaneously across multiple hierarchical metric levels, replacing the augmenting path approach used in prior static algorithms. This enables efficient incremental updates without scanning the entire graph for each arriving request.

2 retrieved papers
Parallelizable algorithm design with CPU and GPU implementations

The algorithm supports parallel request processing where new requests can begin execution while earlier ones are still running. The authors provide efficient implementations on both CPU and GPU, with core subroutines that parallelize naturally, making the algorithm particularly effective for batched-insertion scenarios.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

First constant-factor approximation algorithm for incremental metric bipartite matching with sublinear update time

The authors introduce a deterministic algorithm that maintains an O(1/δ^0.631)-approximate minimum-cost matching in arbitrary metric spaces with amortized insertion time of O(n^(1+δ)). This is the first algorithm for incremental minimum-cost matching that applies to arbitrary metric spaces while guaranteeing provably fast updates.

Contribution

Push-relabel framework for maintaining 1-feasible matchings across hierarchical metric levels

The authors develop a novel push-relabel framework that maintains partial 1-feasible matchings simultaneously across multiple hierarchical metric levels, replacing the augmenting path approach used in prior static algorithms. This enables efficient incremental updates without scanning the entire graph for each arriving request.

Contribution

Parallelizable algorithm design with CPU and GPU implementations

The algorithm supports parallel request processing where new requests can begin execution while earlier ones are still running. The authors provide efficient implementations on both CPU and GPU, with core subroutines that parallelize naturally, making the algorithm particularly effective for batched-insertion scenarios.

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