Algorithms for the Euclidean Bipartite Edge Cover Problem

TL;DR

This paper presents a faster algorithm for solving the Euclidean bipartite edge cover problem in the plane, improving previous solutions and providing approximation algorithms with experimental validation.

Contribution

It introduces an $O(|V|^{2.5} \,\log |V|)$ algorithm for the Euclidean bipartite edge cover problem, surpassing the prior $O(|V|^3)$ solution, and proposes effective approximation methods.

Findings

The new algorithm runs in $O(|V|^{2.5} \,\log |V|)$ time.

Approximation algorithms achieve a 2-approximation ratio.

Experimental results demonstrate practical efficiency.

Abstract

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 $O(|V|^{2.5} \log |V|)$ algorithm based on the Hungarian method using weighted Voronoi diagrams. We also propose some 2-approximation algorithms and give experimental results of our implementations.