A framework for dynamic matching in weighted graphs

Aaron Bernstein, Aditi Dudeja, Zachary Langley

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We introduce a new framework for computing approximate maximum weight matchings. Our primary focus is on the fully dynamic setting, where there is a large gap between the guarantees of the best known algorithms for computing weighted and unweighted matchings. Indeed, almost all current weighted matching algorithms that reduce to the unweighted problem lose a factor of two in the approximation ratio. In contrast, in other sublinear models such as the distributed and streaming models, recent work has largely closed this weighted/unweighted gap. For bipartite graphs, we almost completely settle the gap with a general reduction that converts any algorithm for ?-approximate unweighted matching to an algorithm for (1-)?-approximate weighted matching, while only increasing the update time by an O(logn) factor for constant . We also show that our framework leads to significant improvements for non-bipartite graphs, though not in the form of a universal reduction. In particular, we give two algorithms for weighted non-bipartite matching: 1. A randomized (Las Vegas) fully dynamic algorithm that maintains a (1/2-)-approximate maximum weight matching in worst-case update time O(polylog n) with high probability against an adaptive adversary. Our bounds are essentially the same as those of the unweighted algorithm of Wajc [STOC 2020]. 2. A deterministic fully dynamic algorithm that maintains a (2/3-)-approximate maximum weight matching in amortized update time O(m1/4). Our bounds are essentially the same as those of the unweighted algorithm of Bernstein and Stein [SODA 2016]. A key feature of our framework is that it uses existing algorithms for unweighted matching as black-boxes. As a result, our framework is simple and versatile. Moreover, our framework easily translates to other models, and we use it to derive new results for the weighted matching problem in streaming and communication complexity models.

Original languageEnglish (US)
Title of host publicationSTOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
EditorsSamir Khuller, Virginia Vassilevska Williams
PublisherAssociation for Computing Machinery
Pages668-681
Number of pages14
ISBN (Electronic)9781450380539
DOIs
StatePublished - Jun 15 2021
Event53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy
Duration: Jun 21 2021Jun 25 2021

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/TerritoryItaly
CityVirtual, Online
Period6/21/216/25/21

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Adaptive Adversary
  • Dynamic Algorithms
  • Dynamic Matching
  • Matching Sparsifiers
  • Weighted Matching

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