MAXIMUM CLIQUE and MINIMUM CLIQUE PARTITION in Visibility Graphs

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 maximum clique problem on general graphs implies that no polynomial time algorithm can achieve an approximation ratio of n 1/8− 4 for any > 0, unless NP = P . To demonstrate that allowing holes in the input polygons makes a major difference, we propose an O(n) algorithm for the maximum clique problem on visibility graphs for polygons without holes (other O(n) 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 n 1/14−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.