The Bichromatic Two-Center Problem on Graphs

TL;DR

This paper introduces algorithms for solving the bichromatic two-center problem on graphs and trees, aiming to optimally assign pairs of vertices to two centers to minimize maximum distances, a problem not previously studied on graphs.

Contribution

The paper presents the first algorithms for the bichromatic two-center problem on graphs and trees, including an $O(m^2n ext{log} n ext{log} mn)$ algorithm for general graphs and efficient solutions for trees.

Findings

An $O(m^2n ext{log} n ext{log} mn)$ algorithm for graphs with distance matrix.

An $O(n ext{log} n)$ algorithm for trees.

A linear-time solution for unweighted trees.

Abstract

In this paper, we study the (weighted) bichromatic two-center problem on graphs. The input consists of a graph $G$ of $n$ (weighted) vertices and $m$ edges, and a set $\mathcal{P}$ of pairs of distinct vertices, where no vertex appears in more than one pair. The problem aims to find two points (i.e., centers) on $G$ by assigning vertices of each pair to different centers so as to minimize the maximum (weighted) distance of vertices to their assigned centers (so that the graph can be bi-colored with this goal). To the best of our knowledge, this problem has not been studied on graphs, including tree graphs. In this paper, we propose an $O(m^2n\log n\log mn)$ algorithm for solving the problem on an undirected graph provided with the distance matrix, an $O(n\log n)$-time algorithm for the problem on trees, and a linear-time approach for the unweighted tree version.