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The Bichromatic Two-Center Problem on Graphs

Qi Sun, Jingru Zhang

TL;DR

This work introduces the first algorithms for the weighted bichromatic two-center problem on graphs and trees. It reduces feasibility to a geometric-piercing problem via edge-local centers and distance-function line arrangements, enabling an O(m^2n log n log mn) solution on general graphs and an O(n log n) solution on trees for weighted cases, with a linear-time solution for unweighted trees. The approach hinges on a two-stage framework: compute a global feasibility through local edge-pair tests and solve a piercing problem efficiently with sweep-line data structures. The results extend classic center problems from planar and tree settings to general graphs, offering precise complexity bounds and practical strategies for networked facility-location tasks. The work also highlights open questions, notably whether a linear-time algorithm exists for the vertex-weighted tree case, motivating future exploration of prune-and-search adaptations to bichromatic variants.

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.

The Bichromatic Two-Center Problem on Graphs

TL;DR

This work introduces the first algorithms for the weighted bichromatic two-center problem on graphs and trees. It reduces feasibility to a geometric-piercing problem via edge-local centers and distance-function line arrangements, enabling an O(m^2n log n log mn) solution on general graphs and an O(n log n) solution on trees for weighted cases, with a linear-time solution for unweighted trees. The approach hinges on a two-stage framework: compute a global feasibility through local edge-pair tests and solve a piercing problem efficiently with sweep-line data structures. The results extend classic center problems from planar and tree settings to general graphs, offering precise complexity bounds and practical strategies for networked facility-location tasks. The work also highlights open questions, notably whether a linear-time algorithm exists for the vertex-weighted tree case, motivating future exploration of prune-and-search adaptations to bichromatic variants.

Abstract

In this paper, we study the (weighted) bichromatic two-center problem on graphs. The input consists of a graph of (weighted) vertices and edges, and a set 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 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 algorithm for solving the problem on an undirected graph provided with the distance matrix, an -time algorithm for the problem on trees, and a linear-time approach for the unweighted tree version.

Paper Structure

This paper contains 19 sections, 10 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Our approach.
  4. Outline.
  5. Preliminaries
  6. The problem on a graph
  7. Proof of Lemma \ref{['lem:graphfeasibility']}
  8. Problem reduction.
  9. Solving the geometric piercing problem
  10. Preprocessing.
  11. The algorithm.
  12. Wrapping things up
  13. The problem on a tree
  14. The feasibility test
  15. The algorithm
  16. ...and 4 more sections

Key Result

lemma 1

ref:ChenAn13 Both vertices $v_1(L)$ and $v_2(L)$ can be computed in $O((N+\tau)\log N)$ time, where $\tau$ is the time for a feasibility test.

Figures (4)

  • Figure 1: Illustrate the three cases of function $y = w(v)d(v,x)$ w.r.t. $x$ on edge $e$. Point $x'$ is the (unique) semicircular point of $v$ on $e$.
  • Figure 2: Illustrate that each assignment of pair $P_i=(v_i, u_i)$ leads at most four disjoint rectangles in box $B$ each sharing a vertex of $B$. In (a), $v_i$ (resp., $u_i$) is assigned to center $q_1\in e_1$ (resp., $q_2\in e_2$) so two rectangles $R_1$ and $R_2$ are generated; (b) illustrates the other assignment which leads rectangle $R_3$.
  • Figure 3: Illustrating Observation \ref{['obs:hitting']}. $U_t$ has three (bold) rectangles and $U_{t'}$ has two (non-bold) rectangles. The sweeping line $h$ intersects $B$ at the (blue) line segment $S_j$. The complement segments of $U_t$ and $U_{t'}$ on $S_j$ are $l_t$ and $l_{t'}$.
  • Figure 4: Vertex $v$ contains four split subtrees (marked by dotted lines) and four hanging subtrees (marked by dashed lines); $v$ is a virtual point in its each hanging subtree.

Theorems & Definitions (16)

  • lemma 1
  • lemma 2
  • theorem 1
  • proof
  • lemma 3
  • lemma 4
  • proof
  • lemma 5
  • lemma 6
  • proof
  • ...and 6 more