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 M', and we are asked whether there is a sequence of matchings in G starting with M and ending at M', 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,M'}1. Ito et al. gave a polynomial time algorithm for the problem, which uses the EdmondsGallai decomposition.
In this paper we introduce a natural generalization of the MRP that depends on an integer parameter Δ ≥ 1: here we are allowed to make Δ 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 M' if Δ is sufficiently large, and naturally we would like to minimize Δ. We first devise an optimal transformation procedure for unweighted matching with Δ = 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 worstcase 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 blackbox reduction: Any dynamic algorithm for maintaining a βapproximate 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 worstcase recourse bound O(1/ε). This result generalizes for approximate maximum weight matching, where the update time and worstcase 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 aspectratio. We complement this positive result by showing that, for β = 1+ε, the worstcase recourse bound of any algorithm produced by our reduction is optimal. As a corollary, several key dynamic approximate matching algorithms  with poor worstcase recourse bounds  are strengthened to achieve nearoptimal worstcase recourse bounds with no loss in update time.
BibTeX  Entry
@InProceedings{solomon_et_al:LIPIcs.ITCS.2021.57,
author = {Noam Solomon and Shay Solomon},
title = {{A Generalized Matching Reconfiguration Problem}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {57:157:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771771},
ISSN = {18688969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13596},
URN = {urn:nbn:de:0030drops135965},
doi = {10.4230/LIPIcs.ITCS.2021.57},
annote = {Keywords: Dynamic algorithms, graph matching, reconfiguration problem, recourse bound}
}
Keywords: 

Dynamic algorithms, graph matching, reconfiguration problem, recourse bound 
Collection: 

12th Innovations in Theoretical Computer Science Conference (ITCS 2021) 
Issue Date: 

2021 
Date of publication: 

04.02.2021 