A Faster Algorithm for Maximum Weight Matching on Unrestricted Bipartite Graphs

TL;DR

This paper introduces a new, faster algorithm for solving the maximum weight matching problem in bipartite graphs that supports real-valued weights and improves upon previous time complexity bounds.

Contribution

The paper presents a novel algorithm with improved theoretical time complexity for maximum weight matching in bipartite graphs, supporting real-valued weights without additional constraints.

Findings

Achieves a time complexity of $O( ext{min}(X^3 + E, XE + X^2 ext{log} X))$

Outperforms previous bounds like $O(VE + V^2 ext{log} V)$

Experimental results show average-case complexity of $O(E^{1.4} + LR)$

Abstract

Given a weighted bipartite graph $G = (L, R, E, w)$, the maximum weight matching (MWM) problem seeks to find a matching $M \subseteq E$ that maximizes the total weight $\sum_{e \in M} w(e)$. This paper presents a novel algorithm with a time complexity of $O(\min(X^3 + E, XE + X^2\log X))$, where $X = \min(|L|, |R|)$. Unlike many existing algorithms, our approach supports real-valued weights without additional constraints. Under this condition, our result improves upon the previous best-known bound of $O(VE + V^2\log V)$, or more strictly $O(XE + XV\log V)$, where $V = L \cup R$. The suggested implementation code is simplified and publicly available at https://github.com/ShawxingKwok/Kwok-algorithm, with the average-case time complexity of $O(E^{1.4} + LR)$ estimated from experimental results on random graphs.