On the Ball-Constrained Weighted Maximin Dispersion Problem

TL;DR

This paper introduces a new relaxation and approximation algorithm for the ball-constrained weighted maximin dispersion problem, providing polynomial solutions when m ≤ n and a novel approximation bound in general.

Contribution

It proposes a second-order cone programming relaxation that is tight for m ≤ n and develops a randomized approximation algorithm with a new theoretical bound.

Findings

Relaxation is tight for m ≤ n, enabling polynomial-time solutions.

NP-hardness of the general problem is established.

New approximation bound of (1 - O(√(ln m)/n))/2 for the randomized algorithm.

Abstract

The ball-constrained weighted maximin dispersion problem $(\rm P_{ball})$ is to find a point in an $n$-dimensional Euclidean ball such that the minimum of the weighted Euclidean distance from given $m$ points is maximized. We propose a new second-order cone programming relaxation for $(\rm P_{ball})$. Under the condition $m\le n$, $(\rm P_{ball})$ is polynomial-time solvable since the new relaxation is shown to be tight. In general, we prove that $({\rm P_{ball}})$ is NP-hard. Then, we propose a new randomized approximation algorithm for solving $({\rm P_{ball}})$, which provides a new approximation bound of $\frac{1-O(\sqrt{\ln(m)/n})}{2}$.