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.
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Dumitrescu, A., Jiang, M. (2010). Minimum-Perimeter Intersecting Polygons. In: López-Ortiz, A. (eds) LATIN 2010: Theoretical Informatics. LATIN 2010. Lecture Notes in Computer Science, vol 6034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12200-2_38
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DOI: https://doi.org/10.1007/978-3-642-12200-2_38
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