A Pseudopolynomial Time O (log n )-Approximation Algorithm for Art Gallery Problems

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 3copt. We use a set cover cum VC-dimension based algorithm to solve this restricted problem approximately.

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