A generalization of Hopcroft-Karp algorithm for semi-matchings and covers in bipartite graphs (Maximum semi-matching problem in bipartite graphs)

TL;DR

This paper extends the Hopcroft-Karp algorithm to efficiently find maximum semi-matchings in bipartite graphs with vertex-specific constraints, improving computational complexity for these problems.

Contribution

It introduces a generalized algorithm for maximum $(f,g)$-semi-matching in bipartite graphs with improved running time.

Findings

Achieves a running time of $O(m imes ext{min}( ext{sqrt}( ext{sum of }f(u)), ext{sqrt}( ext{sum of }g(v))))$

Provides a reduction to solve the optimal semi-matching problem in $O( ext{sqrt}(n) m ext{ log } n)$ time

Extends Hopcroft-Karp algorithm to semi-matchings with vertex-specific constraints

Abstract

An $(f,g)$-semi-matching in a bipartite graph $G=(U \cup V,E)$ is a set of edges $M \subseteq E$ such that each vertex $u\in U$ is incident with at most $f(u)$ edges of $M$, and each vertex $v\in V$ is incident with at most $g(v)$ edges of $M$. In this paper we give an algorithm that for a graph with $n$ vertices and $m$ edges, $n\le m$, constructs a maximum $(f,g)$-semi-matching in running time $O(m\cdot \min (\sqrt{\sum_{u\in U}f(u)}, \sqrt{\sum_{v\in V}g(v)}))$. Using the reduction of [5], our result on maximum $(f,g)$-semi-matching problem directly implies an algorithm for the optimal semi-matching problem with running time $O(\sqrt{n}m \log n)$.