A randomized algorithm for finding a maximum clique in the visibility graph of a simple polygon

Abstract : We present a randomized algorithm to compute a clique of maximum size in the visibility graph G of the vertices of a simple polygon P. The input of the problem consists of the visibility graph G, a Hamiltonian cycle describing the boundary of P, and a parameter δ∈(0,1) controlling the probability of error of the algorithm. The algorithm does not require the coordinates of the vertices of P. With probability at least 1-δ the algorithm runs in O( |E(G)|2 / ω(G) log(1/δ)) time and returns a maximum clique, where ω(G) is the number of vertices in a maximum clique in G. A deterministic variant of the algorithm takes O(|E(G)|2) time and always outputs a maximum size clique. This compares well to the best previous algorithm by Ghosh et al. (2007) for the problem, which is deterministic and runs in O(|V(G)|2 |E(G)|) time.
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Discrete Mathematics and Theoretical Computer Science, DMTCS, 2015, Vol. 17 no. 1 (in progress) (1), pp.1--12
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Sergio Cabello, Maria Saumell. A randomized algorithm for finding a maximum clique in the visibility graph of a simple polygon. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2015, Vol. 17 no. 1 (in progress) (1), pp.1--12. 〈hal-01196870〉

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