Dynamic $(1+ε)$-Approximate Matching Size in Truly Sublinear Update Time

TL;DR

This work resolves a central open question in dynamic graph algorithms by giving a fully dynamic algorithm that maintains a $(1+\epsilon)$-approximate size of a maximum matching with update time $m^{0.5-\Omega_{\epsilon}(1)}$ and supports vertex-query access to the current matching. The main technical advance is a first sublinear algorithm for a $(1,\epsilon n)$-approximate maximum matching on dense graphs, achieved via a sublinear matching oracle for induced subgraphs and a template that boosts approximation quality. The results connect dynamic, sublinear, and streaming techniques, introducing two key ingredients: (I) a reduction from $(1,\gamma n)$-approximation to additive-guaranteed subroutines, and (II) LargeMatching oracles for induced subgraphs that circumvent reading cross-edges, enabling sublinear preprocessing and query times. Building on these, the paper then derives a sublinear $(1,\epsilon n)$-approximate matching oracle and, via standard phase-based reduction techniques, a dynamic $(1+\epsilon)$-approximate size maintenance with sublinear update time and near-linear query support. The combination of sublinear subroutines and contraction-based sparsification ultimately yields a polynomial improvement over the naive $O(n)$ bound and advances the understanding of exact value maintenance in dynamic dense graphs.

Abstract

We show a fully dynamic algorithm for maintaining $(1+ε)$-approximate \emph{size} of maximum matching of the graph with $n$ vertices and $m$ edges using $m^{0.5-Ω_ε(1)}$ update time. This is the first polynomial improvement over the long-standing $O(n)$ update time, which can be trivially obtained by periodic recomputation. Thus, we resolve the value version of a major open question of the dynamic graph algorithms literature (see, e.g., [Gupta and Peng FOCS'13], [Bernstein and Stein SODA'16],[Behnezhad and Khanna SODA'22]). Our key technical component is the first sublinear algorithm for $(1,εn)$-approximate maximum matching with sublinear running time on dense graphs. All previous algorithms suffered a multiplicative approximation factor of at least $1.499$ or assumed that the graph has a very small maximum degree.

Theorems & Definitions (72)

• Theorem 1.2
• Theorem 1.3
• Proposition 3.1: Hoeffding bound
• Proposition 3.2: Chernoff bound
• Theorem 4.1
• Lemma 4.2
• Lemma 4.3
• proof : Proof of \ref{['lem:lowdeg oracle']}
• Lemma 4.4
• Lemma 4.5