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arXiv · 2512.08111

The Bichromatic Two-Center Problem on Graphs

Qi Sun·Jingru Zhang
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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.

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Qi Sun, Jingru Zhang. 2025-12-08. The Bichromatic Two-Center Problem on Graphs. https://arxiv.org/abs/2512.08111

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