Approximation Algorithms For The Euclidean Dispersion Problems

TL;DR

This paper introduces improved approximation algorithms for Euclidean dispersion problems, achieving better approximation factors and providing optimal solutions for line placements, advancing the computational methods for spatial point selection.

Contribution

It presents a new polynomial-time approximation algorithm with a factor of 2√3 for the 2-dispersion problem in the plane, and a unified framework for designing such algorithms.

Findings

Achieved a (2√3 + ε)-approximation for 2-dispersion, improving over previous 4√3 factor.

Developed a framework that improves the approximation factor to 2√3 for 2-dispersion.

Provided an optimal solution algorithm for points on a line and a 2-factor approximation for 1-dispersion.

Abstract

In this article, we consider the Euclidean dispersion problems. Let $P=\{p_{1}, p_{2}, \ldots, p_{n}\}$ be a set of $n$ points in $\mathbb{R}^2$. For each point $p \in P$ and $S \subseteq P$, we define $cost_{\gamma}(p,S)$ as the sum of Euclidean distance from $p$ to the nearest $\gamma $ point in $S \setminus \{p\}$. We define $cost_{\gamma}(S)=\min_{p \in S}\{cost_{\gamma}(p,S)\}$ for $S \subseteq P$. In the $\gamma$-dispersion problem, a set $P$ of $n$ points in $\mathbb{R}^2$ and a positive integer $k \in [\gamma+1,n]$ are given. The objective is to find a subset $S\subseteq P$ of size $k$ such that $cost_{\gamma}(S)$ is maximized. We consider both $2$-dispersion and $1$-dispersion problem in $\mathbb{R}^2$. Along with these, we also consider $2$-dispersion problem when points are placed on a line. In this paper, we propose a simple polynomial time $(2\sqrt 3 + \epsilon )$-factor approximation algorithm for the $2$-dispersion problem, for any $\epsilon > 0$, which is an improvement over the best known approximation factor $4\sqrt3$ [Amano, K. and Nakano, S. I., An approximation algorithm for the $2$-dispersion problem, IEICE Transactions on Information and Systems, Vol. 103(3), pp. 506-508, 2020]. Next, we develop a common framework for designing an approximation algorithm for the Euclidean dispersion problem. With this common framework, we improve the approximation factor to $2\sqrt 3$ for the $2$-dispersion problem in $\mathbb{R}^2$. Using the same framework, we propose a polynomial time algorithm, which returns an optimal solution for the $2$-dispersion problem when points are placed on a line. Moreover, to show the effectiveness of the framework, we also propose a $2$-factor approximation algorithm for the $1$-dispersion problem in $\mathbb{R}^2$.