From Unweighted to Weighted Dynamic Matching in Non-Bipartite Graphs: A Low-Loss Reduction

TL;DR

This paper introduces a low-loss reduction transforming approximate maximum cardinality matching algorithms in bipartite graphs into approximate maximum weight matching algorithms in general graphs, significantly improving dynamic matching solutions.

Contribution

It presents the first low-loss reduction from unweighted to weighted dynamic matching in non-bipartite graphs, closing the approximation gap with only polylogarithmic overhead.

Findings

Achieves near-optimal dynamic MWM algorithms with low update time overhead.

Provides the first conditional lower bound for approximate partially dynamic matching.

Introduces a new primal-dual framework for reduction to bipartite induced matching queries.

Abstract

We study the approximate maximum weight matching (MWM) problem in a fully dynamic graph subject to edge insertions and deletions. We design meta-algorithms that reduce the problem to the unweighted approximate maximum cardinality matching (MCM) problem. Despite recent progress on bipartite graphs -- Bernstein-Dudeja-Langley (STOC 2021) and Bernstein-Chen-Dudeja-Langley-Sidford-Tu (SODA 2025) -- the only previous meta-algorithm that applied to non-bipartite graphs suffered a $\frac{1}{2}$ approximation loss (Stubbs-Williams, ITCS 2017). We significantly close the weighted-and-unweighted gap by showing the first low-loss reduction that transforms any fully dynamic $(1-\varepsilon)$-approximate MCM algorithm on bipartite graphs into a fully dynamic $(1-\varepsilon)$-approximate MWM algorithm on general (not necessarily bipartite) graphs, with only a $\mathrm{poly}(\log n/\varepsilon)$ overhead in the update time. Central to our approach is a new primal-dual framework that reduces the computation of an approximate MWM in general graphs to a sequence of approximate induced matching queries on an auxiliary bipartite extension. In addition, we give the first conditional lower bound on approximate partially dynamic matching with worst-case update time.