Optimum matchings in weighted bipartite graphs

TL;DR

This paper presents efficient algorithms for identifying edges in minimum weight matchings and enumerating all such matchings in weighted bipartite graphs, utilizing linear programming duality and graph reduction techniques.

Contribution

It introduces a novel approach to reduce weighted matching problems to unweighted variants using an $ ext{epsilon}$-optimal dual solution, enabling faster algorithms.

Findings

Linear time algorithm for finding edges in minimum weight perfect matchings.

An $O(\sqrt{n}m ext{log}(nW))$ time algorithm for the problem.

Effective reduction of weighted problems to unweighted variants.

Abstract

Given an integer weighted bipartite graph $\{G=(U\sqcup V, E), w:E\rightarrow \mathbb{Z}\}$ we consider the problems of finding all the edges that occur in some minimum weight matching of maximum cardinality and enumerating all the minimum weight perfect matchings. Moreover, we construct a subgraph $G_{cs}$ of $G$ which depends on an $\epsilon$-optimal solution of the dual linear program associated to the assignment problem on $\{G,w\}$ that allows us to reduced this problems to their unweighed variants on $G_{cs}$. For instance, when $G$ has a perfect matching and we have an $\epsilon$-optimal solution of the dual linear program associated to the assignment problem on $\{G,w\}$, we solve the problem of finding all the edges that occur in some minimum weight perfect matching in linear time on the number of edges. Therefore, starting from scratch we get an algorithm that solves this problem in time $O(\sqrt{n}m\log(nW))$, where $n=|U|\geq |V|$, $m=|E|$, and $W={\rm max}\{|w(e)|\, :\, e\in E\}$.