Minimum-Perimeter Intersecting Polygons

Abstract

Given a set ${\mathcal S}$ of segments in the plane, a polygon P is an intersecting polygon of ${\mathcal S}$ if every segment in ${\mathcal S}$ intersects the interior or the boundary of P. The problem MPIP of computing a minimum-perimeter intersecting polygon of a given set of n segments in the plane was first considered by Rappaport in 1995. This problem is not known to be polynomial, nor it is known to be NP-hard. Rappaport (1995) gave an exponential-time exact algorithm for MPIP . Hassanzadeh and Rappaport (2009) gave a polynomial-time approximation algorithm with ratio $\frac{\pi}{2} \approx 1.58$. In this paper, we present two improved approximation algorithms for MPIP: a 1.28-approximation algorithm by linear programming, and a polynomial-time approximation scheme by discretization and enumeration. Our algorithms can be generalized for computing an approximate minimum-perimeter intersecting polygon of a set of convex polygons in the plane. From the other direction, we show that computing a minimum-perimeter intersecting polygon of a set of (not necessarily convex) simple polygons is NP-hard.