Algorithms for the Euclidean Bipartite Edge Cover Problem
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Given a graph G=(V,E) with costs on its edges, the minimum-cost edge cover problem consists of finding a subset of E covering all vertices in V at minimum cost. If G is bipartite, this problem can be solved in time O(|V|^3) via a well-known reduction to a maximum-cost matching problem on G. If in addition V is a set of points on the Euclidean line, Collanino et al. showed that the problem can be solved in time O(|V| log |V|) and asked whether it can be solved in time o(|V|^3) if V is a set of points on the Euclidean plane. We answer this in the affirmative, giving an O(|V|^2.5log |V|) algorithm based on the Hungarian method using weighted Voronoi diagrams. We also propose some 2-approximation algorithms and give experimental results of our implementations.
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