Faster Semi-Streaming and Dynamic Approximate Maximum Weight Matchings in General Graphs: A Unified Framework

Abstract

We study the $(1-\varepsilon)$-approximate maximum weight matching problem in general graphs with $n$ vertices and integer edge weights in $[1,W]$. For $W=n^{O(1)}$, we obtain faster algorithms in the semi-streaming and dynamic settings: (1) Semi-streaming: Deterministic $O(\varepsilon^{-2})$-pass and $O(\varepsilon^{-2}\log(1/\varepsilon))$-pass algorithms for unweighted and weighted matching. (2) Incremental: A deterministic algorithm for weighted matching processing $m$ edge insertions in total time $O((m\varepsilon^{-3}+n\varepsilon^{-6})\log^3 n)$. (3) Decremental: A randomized algorithm for weighted matching processing all edge deletions from an $m$-edge graph in total time $O((m\varepsilon^{-8}+n\varepsilon^{-47})\log^{O(1)}n)$ with high probability. (4) Fully dynamic: A deterministic reduction from weighted matching in general graphs to unweighted matching in bipartite graphs incurring $O(\varepsilon^{-4}\log^{O(1)}(1/\varepsilon))$ multiplicative update-time overhead, independent of $n$. Underlying these results is a unified framework that reduces the problem to repeatedly solving two simpler matching subproblems: a $(1-\varepsilon)$-approximate maximum weight matching on a sparse subgraph with $O(n/\varepsilon)$ edges, and a constant approximation on the remaining edges. The framework builds on dual entropy regularization, a smoothing technique underlying efficient solvers for packing linear programs. To handle the exponentially many odd-set constraints, we adapt the smoothing technique to regularize only the vertex duals and develop a new cut partition that preserves odd-set feasibility during optimization.