Paper by Erik D. Demaine

Reference:
Ajay Deshpande, Taejung Kim, Erik D. Demaine, and Sanjay E. Sarma, “A Pseudopolynomial Time O(log n)-Approximation Algorithm for Art Gallery Problems”, in Proceedings of the 10th Workshop on Algorithms and Data Structures (WADS 2007), Lecture Notes in Computer Science, volume 4619, Halifax, Nova Scotia, Canada, August 15–17, 2007, pages 163–174.

Abstract:
In this paper, we give a O(log copt)-approximation algorithm for the point guard problem where copt is the optimal number of guards. Our algorithm runs in time polynomial in n, the number of walls of the art gallery and the spread Δ, which is defined as the ratio between the longest and shortest pairwise distances. Our algorithm is pseudopolynomial in the sense that it is polynomial in the spread Δ as opposed to polylogarithmic in the spread Δ, which could be exponential in the number of bits required to represent the vertex positions. The special subdivision procedure in our algorithm finds a finite set of potential guard-locations such that the optimal solution to the art gallery problem where guards are restricted to this set is at most 3 copt. We use a set cover cum VC-dimension based algorithm to solve this restricted problem approximately.

Updates:
Unfortunately, the main result of this paper is flawed. In 2016, Édouard Bonnet and Tillman Miltzow showed that the goal of our Step 2 is in fact impossible, but that similar approaches lead to an approximation algorithm.

Length:
The paper is 12 pages.

Availability:
The paper is available in PDF (151k).
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Last updated July 21, 2017 by Erik Demaine.