TY - GEN
T1 - A generalized matching reconfiguration problem
AU - Solomon, Noam
AU - Solomon, Shay
N1 - Publisher Copyright:
© Noam Solomon and Shay Solomon.
PY - 2021/2/1
Y1 - 2021/2/1
N2 - 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.
AB - 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.
KW - Dynamic algorithms
KW - Graph matching
KW - Reconfiguration problem
KW - Recourse bound
UR - http://www.scopus.com/inward/record.url?scp=85115269967&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ITCS.2021.57
DO - 10.4230/LIPIcs.ITCS.2021.57
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AN - SCOPUS:85115269967
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021
A2 - Lee, James R.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 12th Innovations in Theoretical Computer Science Conference, ITCS 2021
Y2 - 6 January 2021 through 8 January 2021
ER -