A generalization of Hopcroft-Karp algorithm for semi-matchings and covers in bipartite graphs

Details A generalization of Hopcroft-Karp algorithm for semi-matchings and covers in bipartite graphs A generalization of Hopcroft-Karp algorithm for semi-matchings and covers in bipartite graphs

2011-03-06

arxiv.org/abs/1103.1091v1

Computer Science

Data Structures and Algorithms

Scientific paper

The aim of the paper is to study a generalized problem of the classical matching problem in bipartite graphs. A {\em semi-matching} in a bipartite graph $G = (U \cup V, E)$ is a set of edges $M \subseteq E$ such that each vertex in $U$ is incident to at most one edge in $M$. An $(f,g)$-cover 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 from $M$, and each vertex $v\in V$ in incident with at most $g(v)$ edges from $M$. We provide an algorithm which finds a maximum semi-matching in running time $O(\sqrt{n} m)$ and solves the general $(f,g)$-cover in running time $O(m^{3/2})$. Semi-matchings and $(f,g)$-covers have various applications in load balancing and routing problems.

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