The Connected k-Vertex One-Center Problem on Graphs

TL;DR

This paper introduces algorithms for a generalized connected k-vertex one-center problem on graphs, optimizing the placement of a point to minimize maximum distances to k connected vertices, with solutions for weighted, unweighted, and tree graphs.

Contribution

It presents the first algorithms for the generalized problem, extending the classical one-center problem to cases where only k vertices are considered, with different complexities for weighted, unweighted, and tree graphs.

Findings

Weighted case algorithm runs in $O(mn\log n\log mn + m^2\log n\log mn)$ time.

Unweighted case algorithm runs in $O(mn\log n)$ time with distance matrix.

Tree graph algorithms achieve $O(n\log^2 n\log k)$ and $O(n\log^2 n)$ time complexities for weighted and unweighted cases, respectively.

Abstract

We consider a generalized version of the (weighted) one-center problem on graphs. Given an undirected graph $G$ of $n$ vertices and $m$ edges and a positive integer $k\leq n$, the problem aims to find a point in $G$ so that the maximum (weighted) distance from it to $k$ connected vertices in its shortest path tree(s) is minimized. No previous work has been proposed for this problem except for the case $k=n$, that is, the classical graph one-center problem. In this paper, an $O(mn\log n\log mn + m^2\log n\log mn)$-time algorithm is proposed for the weighted case, and an $O(mn\log n)$-time algorithm is presented for the unweighted case, provided that the distance matrix for $G$ is given. When $G$ is a tree graph, we propose an algorithm that solves the weighted case in $O(n\log^2 n\log k)$ time with no given distance matrix, and improve it to $O(n\log^2 n)$ for the unweighted case.