Fast Parallel Algorithms for the Clique Separator Decomposition

We give an efficient NC algorithm for finding a clique separator decomposition of an arbitrary graph, that is, a series of cliques whose removal disconnects the graph. This algorithm allows one to extend a large body of results which were originally formulated for chordal graphs to other classes of graphs. Our algorithm is optimal to within a polylogarithmic factor of Tarjan's O(mn) time sequential algorithm. The decomposition can also be used to find NC algorithms for some optimization problems on special families of graphs, assuming these problems can be solved in NC for the prime graphs of the decomposition. These optimization problems include: finding a maximum-weight clique, a minimum coloring, a maximum-weight independent set, and a minimum fill-in elimination order. We also give the first parallel algorithms for solving these problems by using the clique separator decomposition. Our maximum-weight independent set algorithm applied to chordal graphs yields the most efficient know..