Decremental Matching in General Graphs

TL;DR

This paper presents a randomized decremental algorithm for maintaining an approximate maximum matching in general graphs with constant update time, overcoming previous barriers and matching the efficiency of incremental algorithms.

Contribution

It introduces the first $O_{ ext{ε}}(1)$ update time decremental algorithm for approximate maximum matching in general graphs, closing the gap with bipartite graph results.

Findings

Achieves $O_{ ext{ε}}(1)$ update time for decremental matching in general graphs.

Works against an adaptive adversary.

Completes the picture for partially dynamic matching algorithms.

Abstract

We consider the problem of maintaining an approximate maximum integral matching in a dynamic graph $G$, while the adversary makes changes to the edges of the graph. The goal is to maintain a $(1+\epsilon)$-approximate maximum matching for constant $\epsilon>0$, while minimizing the update time. In the fully dynamic setting, where both edge insertion and deletions are allowed, Gupta and Peng (see \cite{GP13}) gave an algorithm for this problem with an update time of $O(\sqrt{m}/\epsilon^2)$. Motivated by the fact that the $O_{\epsilon}(\sqrt{m})$ barrier is hard to overcome (see Henzinger, Krinninger, Nanongkai, and Saranurak [HKNS15]); Kopelowitz, Pettie, and Porat [KPP16]), we study this problem in the \emph{decremental} model, where the adversary is only allowed to delete edges. Recently, Bernstein, Probst-Gutenberg, and Saranurak (see [BPT20]) gave an $O_{\epsilon}(1)$ update time decremental algorithm for this problem in \emph{bipartite graphs}. However, beating $O(\sqrt{m})$ update time remained an open problem for \emph{general graphs}. In this paper, we bridge the gap between bipartite and general graphs, by giving an $O_{\epsilon}(1)$ update time algorithm that maintains a $(1+\epsilon)$-approximate maximum integral matching under adversarial deletions. Our algorithm is randomized, but works against an adaptive adversary. Together with the work of Grandoni, Leonardi, Sankowski, Schwiegelshohn, and Solomon [GLSSS19] who give an $O_{\epsilon}(1)$ update time algorithm for general graphs in the \emph{incremental} (insertion-only) model, our result essentially completes the picture for partially dynamic matching.