Scaling Algorithms for Weighted Matching in General Graphs

TL;DR

This paper introduces a new scaling algorithm for weighted perfect matching in general graphs, achieving improved running time over previous methods and matching the efficiency of algorithms for unweighted matchings.

Contribution

The paper presents a novel scaling algorithm that significantly improves the time complexity for weighted matching in general graphs, surpassing a 25-year-old prior algorithm.

Findings

Runs in $O(m ext{sqrt}(n) ext{log}(nN))$ time

Matches the efficiency of unweighted matching algorithms on sparse graphs

Improves upon the previous $O(m ext{sqrt}(n ext{log} n ext{alpha}(m,n)) ext{log}(nN))$ algorithm

Abstract

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in $O(m\sqrt{n}\log(nN))$ time, $O(m\sqrt{n})$ per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs. Here $m,n,$ and $N$ bound the number of edges, vertices, and magnitude of any edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in $O(m\sqrt{n\log n\alpha(m,n)} \log(nN))$ time.