Exact algorithms for terrain guarding

Pradeesha Ashok, Fedor V. Fomin, Sudeshna Kolay, Saket Saurabh, Meirav Zehavi

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

Given a 1.5-dimensional terrain T, also known as an x-monotone polygonal chain, the Terrain Guarding problem seeks a set of points of minimum size onT that guards all of the points onT. Here, we say that a point p guards a point q if no point of the line segment pq is strictly below T. The Terrain Guarding problem has been extensively studied for over 20 years. In 2005 it was already established that this problem admits a constant-factor approximation algorithm (SODA 2005). However, only in 2010 King and Krohn (SODA 2010) finally showed that Terrain Guarding is NP-hard. In spite of the remarkable developments in approximation algorithms for Terrain Guarding, next to nothing is known about its parameterized complexity. In particular, the most intriguing open questions in this direction ask whether, if parameterized by the size k of a solution guard set, it admits a subexponential-time algorithm and whether it is fixed-parameter tractable. In this article, we answer the first question affirmatively by developing an nO(k)-time algorithm for both Discrete Terrain Guarding and Continuous Terrain Guarding. We also make non-trivial progress with respect to the second question: we show that Discrete Orthogonal Terrain Guarding, a well-studied special case of Terrain Guarding, is fixed-parameter tractable.

Original languageEnglish
Article number25
JournalACM Transactions on Algorithms
Volume14
Issue number2
DOIs
StatePublished - 1 Jun 2018

Keywords

  • Art gallery
  • Exponential-time algorithms
  • Terrain guarding

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