On load-balanced semi-matchings for weighted bipartite graphs

Authors:
Chor Ping Low
Affiliations:
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore
Venue:
TAMC'06 Proceedings of the Third international conference on Theory and Applications of Models of Computation
Year:
2006

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Abstract

A semi-matching on a bipartite graph G=(U ∪ V, E) is a set of edges X⊆E such that each vertex in U is incident to exactly one edge in X. The sum of the weights of the vertices from U that are assigned (semi-matched) to some vertex v ∈ V is referred to as the load of vertex v. In this paper, we consider the problem to finding a semi-matching that minimizes the maximum load among all vertices in V. This problem has been shown to be solvable in polynomial time by Harvey et. al [3] and Fakcharoenphol et. al [5] for unweighted graphs. However, the computational complexity for the weighted version of the problem was left as an open problem. In this paper, we prove that the problem of finding a semi-matching that minimizes the maximum load among all vertices in a weighted bipartite graph is NP-complete. A $\frac{3}{2}$-approximation algorithm is proposed for this problem.