License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.49
URN: urn:nbn:de:0030-drops-59412
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5941/
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Kostitsyna, Irina ; Löffler, Maarten ; Polishchuk, Valentin ; Staals, Frank

On the Complexity of Minimum-Link Path Problems

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LIPIcs-SoCG-2016-49.pdf (0.6 MB)

Abstract

We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the min-link path's vertices or edges can be restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2D, and provide first results in dimensions 3 and higher for several versions of the problem.

Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [Ghosh et al. 2012] despite a large body of work on the topic. We also resolve the open problem from [Mitchell et al. 1992] mentioned in the handbook [Goodman and O'Rourke, 2004] (see Chapter 27.5, Open problem 3) and The Open Problems Project [Demaine et al. TOPP] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.

BibTeX - Entry

@InProceedings{kostitsyna_et_al:LIPIcs:2016:5941,
  author =	{Irina Kostitsyna and Maarten L{\"o}ffler and Valentin Polishchuk and Frank Staals},
  title =	{{On the Complexity of Minimum-Link Path Problems}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{49:1--49:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5941},
  URN =		{urn:nbn:de:0030-drops-59412},
  doi =		{10.4230/LIPIcs.SoCG.2016.49},
  annote =	{Keywords: minimum-linkpath, diffuse reflection, terrain, bit complexity, NP-hardness}
}

Keywords: minimum-linkpath, diffuse reflection, terrain, bit complexity, NP-hardness
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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