Computing Smallest Convex Intersecting Polygons

TL;DR

This paper develops approximation algorithms and exact solutions for finding the smallest convex polygon intersecting a set of objects, including convex polygons and segments, optimizing perimeter and area.

Contribution

It introduces an FPTAS for minimum-perimeter and minimum-area convex intersecting polygons for convex objects, and an exact polynomial-time algorithm for segments.

Findings

FPTAS for convex polygons with total complexity n

Exact polynomial algorithm for segments

Extends known algorithms beyond lines and disjoint segments

Abstract

A polygon C is an intersecting polygon for a set O of objects in the plane if C intersects each object in O, where the polygon includes its interior. We study the problem of computing the minimum-perimeter intersecting polygon and the minimum-area convex intersecting polygon for a given set O of objects. We present an FPTAS for both problems for the case where O is a set of possibly intersecting convex polygons in the plane of total complexity n. Furthermore, we present an exact polynomial-time algorithm for the minimum-perimeter intersecting polygon for the case where O is a set of n possibly intersecting segments in the plane. So far, polynomial-time exact algorithms were only known for the minimum perimeter intersecting polygon of lines or of disjoint segments.