Beating the Folklore Algorithm for Dynamic Matching

TL;DR

This paper introduces the first deterministic algorithms that outperform the folklore approach for dynamic graph matching, achieving better approximation ratios and worst-case update times in both unweighted and weighted cases.

Contribution

It presents new deterministic algorithms for dynamic matching that surpass the folklore algorithm in approximation ratio and update time, including characterizations of kernels and improved recourse techniques.

Findings

Achieved a (2−Ω(1))-approximate algorithm with O(n^{3/4}) update time.

Developed a (2+ε)-approximate weighted matching algorithm with O(√n) update time.

Characterized tight cases for kernels, enhancing understanding of approximation barriers.

Abstract

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {\em both} approximation ratio and worst-case update time. Specifically, we give a $(2-\Omega(1))$-approximate algorithm with $O(m^{3/8})=O(n^{3/4})$ worst-case update time in $n$-node, $m$-edge graphs. For sufficiently small constant $\epsilon>0$, no deterministic $(2+\epsilon)$-approximate algorithm with worst-case update time $O(n^{0.99})$ was known. Our second result is the first deterministic $(2+\epsilon)$-approximate weighted matching algorithm with $O_\epsilon(1)\cdot O(\sqrt[4]{m}) = O_\epsilon(1)\cdot O(\sqrt{n})$ worst-case update time. Our main technical contributions are threefold: first, we characterize the tight cases for \emph{kernels}, which are the well-studied matching sparsifiers underlying much of the $(2+\epsilon)$-approximate dynamic matching literature. This characterization, together with multiple ideas -- old and new -- underlies our result for breaking the approximation barrier of $2$. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the \emph{recourse} of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural $\frac{3}{2}$ factor in the approximation ratio which such approaches naturally incur.