A Faster Algorithm for Maximum Weight Matching on Unrestricted Bipartite Graphs

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 $X=\min (\mathrm{\mid }L\mathrm{\mid },\mathrm{\mid }R\mathrm{\mid })$. 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.