Incremental $(1-\epsilon)$-approximate dynamic matching in $O(poly(1/\epsilon))$ update time

TL;DR

This paper introduces a simple, deterministic algorithm for maintaining a near-optimal matching in a dynamic bipartite graph with efficient update times, handling both insertions and deletions.

Contribution

It presents the first deterministic $(1- ext{epsilon})$-approximate dynamic matching algorithm with polynomial update time for edge insertions, extending to handle vertex deletions.

Findings

Achieves $O(poly(1/epsilon))$ amortized update time.

Handles both edge insertions and vertex deletions.

Simpler than previous algorithms with comparable guarantees.

Abstract

In the dynamic approximate maximum bipartite matching problem we are given bipartite graph $G$ undergoing updates and our goal is to maintain a matching of $G$ which is large compared the maximum matching size $\mu(G)$. We define a dynamic matching algorithm to be $\alpha$ (respectively $(\alpha, \beta)$)-approximate if it maintains matching $M$ such that at all times $|M | \geq \mu(G) \cdot \alpha$ (respectively $|M| \geq \mu(G) \cdot \alpha - \beta$). We present the first deterministic $(1-\epsilon )$-approximate dynamic matching algorithm with $O(poly(\epsilon ^{-1}))$ amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or exponential in $1/\epsilon $ [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive $(1, \epsilon \cdot n)$-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for $(1-\epsilon )$-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of $G$ in a fully dynamic manner.