Max-Min $k$-Dispersion on a Convex Polygon

TL;DR

This paper introduces an exact fixed-parameter algorithm for the max-min $k$-dispersion problem on convex polygons, improving efficiency for small $k$, and also provides a fast approximation for the case when $k=3$.

Contribution

It presents a new fixed-parameter algorithm with better performance for small $k$ and a rapid approximation algorithm for $k=3$ in convex position.

Findings

Exact algorithm runs in $O(2^k(n^2 ext{log} n + n( ext{log}^2 n)( ext{log} k)))$ time.

Improves upon previous algorithms when $k<c ext{log}^2 n$.

Provides a $O( ext{log} n)$-time $rac{1}{2 ext{sqrt}2}$-approximation for $k=3$.

Abstract

In this paper, we consider the following $k$-dispersion problem. Given a set $S$ of $n$ points placed in the plane in a convex position, and an integer $k$ ($0<k<n$), the objective is to compute a subset $S'\subset S$ such that $|S'|=k$ and the minimum distance between a pair of points in $S'$ is maximized. Based on the bounded search tree method we propose an exact fixed-parameter algorithm in $O(2^k(n^2\log n+n(\log^2 n)(\log k)))$ time, for this problem, where $k$ is the parameter. The proposed exact algorithm is better than the current best exact exponential algorithm [$n^{O(\sqrt{k})}$-time algorithm by Akagi et al.,(2018)] whenever $k<c\log^2{n}$ for some constant $c$. We then present an $O(\log{n})$-time $\frac{1}{2\sqrt{2}}$-approximation algorithm for the problem when $k=3$ if the points are given in convex position order.