A generalized matching reconfiguration problem

Noam Solomon, Shay Solomon

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

7 Scopus citations

Abstract

The goal in reconfiguration problems is to compute a gradual transformation between two feasible solutions of a problem such that all intermediate solutions are also feasible. In the Matching Reconfiguration Problem (MRP), proposed in a pioneering work by Ito et al. from 2008, we are given a graph G and two matchings M and M0, and we are asked whether there is a sequence of matchings in G starting with M and ending at M0, each resulting from the previous one by either adding or deleting a single edge in G, without ever going through a matching of size < min{|M|, |M0|}-1. Ito et al. gave a polynomial time algorithm for the problem, which uses the Edmonds-Gallai decomposition. In this paper we introduce a natural generalization of the MRP that depends on an integer parameter ∆ ≥ 1: here we are allowed to make ∆ changes to the current solution rather than 1 at each step of the transformation procedure. There is always a valid sequence of matchings transforming M to M0 if ∆ is sufficiently large, and naturally we would like to minimize ∆. We first devise an optimal transformation procedure for unweighted matching with ∆ = 3, and then extend it to weighted matchings to achieve asymptotically optimal guarantees. The running time of these procedures is linear. We further demonstrate the applicability of this generalized problem to dynamic graph matchings. In this area, the number of changes to the maintained matching per update step (the recourse bound) is an important quality measure. Nevertheless, the worst-case recourse bounds of almost all known dynamic matching algorithms are prohibitively large, much larger than the corresponding update times. We fill in this gap via a surprisingly simple black-box reduction: Any dynamic algorithm for maintaining a β-approximate maximum cardinality matching with update time T, for any β ≥ 1, T and ε > 0, can be transformed into an algorithm for maintaining a (β(1 + ε))-approximate maximum cardinality matching with update time T + O(1/ε) and worst-case recourse bound O(1/ε). This result generalizes for approximate maximum weight matching, where the update time and worst-case recourse bound grow from T + O(1/ε) and O(1/ε) to T + O(ψ/ε) and O(ψ/ε), respectively; ψ is the graph aspect-ratio. We complement this positive result by showing that, for β = 1 + ε, the worst-case recourse bound of any algorithm produced by our reduction is optimal. As a corollary, several key dynamic approximate matching algorithms – with poor worst-case recourse bounds – are strengthened to achieve near-optimal worst-case recourse bounds with no loss in update time.

Original languageEnglish
Title of host publication12th Innovations in Theoretical Computer Science Conference, ITCS 2021
EditorsJames R. Lee
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771771
DOIs
StatePublished - 1 Feb 2021
Event12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online
Duration: 6 Jan 20218 Jan 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume185
ISSN (Print)1868-8969

Conference

Conference12th Innovations in Theoretical Computer Science Conference, ITCS 2021
CityVirtual, Online
Period6/01/218/01/21

Funding

FundersFunder number
Blavatnik Family Foundation
Israel Science Foundation1991/19

    Keywords

    • Dynamic algorithms
    • Graph matching
    • Reconfiguration problem
    • Recourse bound

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