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Deterministic Rounding of Dynamic Fractional Matchings

Authors Sayan Bhattacharya, Peter Kiss

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Sayan Bhattacharya
  • Department of Computer Science, University of Warwick, Coventry, UK
Peter Kiss
  • Department of Computer Science, University of Warwick, Coventry, UK

Funding

  • Bhattacharya, Sayan: Supported by EPSRC Grant EP/S03353X/1.

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Sayan Bhattacharya and Peter Kiss. Deterministic Rounding of Dynamic Fractional Matchings. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.27

Abstract

We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a (2-δ)-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a (1+δ)-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintaining a (2+δ)-approximate maximum matching in a fully dynamic general graph, in small polylogarithmic (specifically, O(log⁴ n)) update time. These results are respectively obtained by applying our framework on top of the fractional matching algorithms of Bhattacharya et al. [STOC'16], Bernstein et al. [FOCS'20], and Bhattacharya and Kulkarni [SODA'19]. 
Previously, there were two known general-purpose rounding schemes for dynamic fractional matchings. Both these schemes, by Arar et al. [ICALP'18] and Wajc [STOC'20], were randomized. 
Our rounding scheme works by maintaining a good matching-sparsifier with bounded arboricity, and then applying the algorithm of Peleg and Solomon [SODA'16] to maintain a near-optimal matching in this low arboricity graph. To the best of our knowledge, this is the first dynamic matching algorithm that works on general graphs by using an algorithm for low-arboricity graphs as a black-box subroutine. This feature of our rounding scheme might be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Matching
  • Dynamic Algorithms
  • Data Structures

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