Maximum Clique and Minimum Clique Partition in Visibility Graphs

Abstract

In an alternative approach to “characterizing” the graph class of visibility graphs of simple polygons, we study the problem of finding a maximum clique in the visibility graph of a simple polygon with n vertices. We show that this problem is very hard, if the input polygons are allowed to contain holes: a gap-preserving reduction from the maxi-mum clique problem on general graphs implies that no polynomial time algorithm can achieve an approximation ratio of \( \frac{{n^{1/8 - \in } }} {4} \) for any ∈ > 0, unless NP = P. To demonstrate that allowing holes in the input polygons makes a major difference, we propose an O(n3) algorithm for the maximum clique problem on visibility graphs for polygons without holes (other O(n 3) algorithms for this problem are already known [3,6,7]). Our algorithm also finds the maximum weight clique, if the polygon vertices are weighted.We then proceed to study the problem of partitioning the vertices of a visibility graph of a polygon into a minimum number of cliques. This problem is APX-hard for polygons without holes (i.e., there exists a constant γ > 0 such that no polynomial time algorithm can achieve an approximation ratio of 1 +γ ). We present an approximation algorithm for the problem that achieves a logarithmic approximation ratio by iteratively applying the algorithm for finding maximum weighted cliques. Finally, we show that the problem of partitioning the vertices of a visibility graph of a polygon with holes cannot be approximated with a ratio of n1¼ -γ/4 for any γ > 0 by proposing a gap-preserving reduction. Thus, the presence of holes in the input polygons makes this partitioning problem provably harder.