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Parameterized Hardness of Art Gallery Problems

Authors Édouard Bonnet, Tillmann Miltzow

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Keywords
  • art gallery problem
  • computational geometry
  • parameterized complexity
  • ETH-based lower bound
  • geometric set cover/hitting set

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Abstract

Given a simple polygon P on n vertices, two points x,y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S.
The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard.   For both variants, we rule out a f(k)*n^{o(k/log k)} algorithm, for any computable function f, where k := |S| is the number of guards, unless the Exponential Time Hypothesis fails. These lower bounds almost match the n^{O(k)} algorithms that exist for both problems.

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Édouard Bonnet and Tillmann Miltzow. Parameterized Hardness of Art Gallery Problems. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ESA.2016.19

Author Details

Édouard Bonnet
Tillmann Miltzow

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