How to Cover a Point Set with a V-Shape of Minimum Width

TL;DR

This paper introduces algorithms for efficiently computing the minimum-width balanced V-shape covering a point set, including an exact algorithm, a PTAS for approximate solutions, and a simple constant-factor approximation.

Contribution

It presents the first exact algorithm and a PTAS for the minimum-width balanced V-shape covering problem, along with a simple approximation method.

Findings

Exact algorithm runs in O(n^2 log n) time.

PTAS provides (1+epsilon)-approximate solutions efficiently.

A simple constant-factor approximation algorithm is also introduced.

Abstract

A balanced V-shape is a polygonal region in the plane contained in the union of two crossing equal-width strips. It is delimited by two pairs of parallel rays that emanate from two points x, y, are contained in the strip boundaries, and are mirror-symmetric with respect to the line xy. The width of a balanced V-shape is the width of the strips. We first present an O(n^2 log n) time algorithm to compute, given a set of n points P, a minimum-width balanced V-shape covering P. We then describe a PTAS for computing a (1+epsilon)-approximation of this V-shape in time O((n/epsilon)log n+(n/epsilon^(3/2))log^2(1/epsilon)). A much simpler constant-factor approximation algorithm is also described.