From Unweighted to Weighted Dynamic Matching in Non-Bipartite Graphs: A Low-Loss Reduction
Aaron Bernstein, Jiale Chen
TL;DR
The paper tackles dynamic maximum-weight matching (MWM) on general graphs, focusing on the fully dynamic setting. It introduces a low-loss meta-algorithm that reduces fully dynamic MWM to fully dynamic unweighted MCM by leveraging a primal-dual framework and a bipartite induced-matching oracle, achieving a $(1-\\varepsilon)$-approximation with poly$(\\log n/\\varepsilon)$ update-time overhead. The approach bridges the weighted/unweighted gap for non-bipartite graphs and connects to conditional lower bounds in the partially dynamic regime, while also delivering offline and sublinear-time implications under certain graph-density assumptions (e.g., ORS graphs). Overall, the work provides a structured, oracle-based reduction that enables near-optimal dynamic MWM in general graphs by composition with state-of-the-art MCM machinery, and it establishes conditional limits and avenues for future sublinear-time advances. The results have potential practical impact on dynamic networks where maintaining high-weight resource allocations under updates is critical, and they lay groundwork for cross-pollination with streaming, parallel, and distributed models via the primal-dual framework.
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.