Minimum-perimeter intersecting polygons

  • Authors:

  • Adrian Dumitrescu;Minghui Jiang

  • Affiliations:

  • Department of Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI;Department of Computer Science, Utah State University, Logan, UT

  • Venue:
  • LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
  • Year:
  • 2010

  • Opaque sets

    APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques

Quantified Score

Hi-index 0.00

Visualization

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.