Revisiting Decomposition by Clique Separators.

We study the complexity of decomposing a graph by means of clique separators. This common algorithmic tool, first introduced by Tarjan, allows one to cut a graph into smaller pieces, and so it can be applied to preprocess the graph in the computation of optimization problems. However, the best-known algorithms for computing a decomposition have respective O(nm)-time and O(n((3+alpha)/2)) = o(n(2.69))-time complexity with alpha < 2.3729 being the exponent for matrix multiplication. Such running times are prohibitive for large graphs. Here we prove that for every graph G, a decomposition can be computed in O(T(G) + min{n(alpha), omega(2)n})-time with T(G) and omega being, respectively, the time needed to compute a minimal triangulation of G and the clique-number of G. In particular, it implies that every graph can be decomposed by clique separators in O(n(alpha) log n)-time. Based on prior work from Kratsch et al., we prove in addition that decomposing a graph by clique-separators is as least as hard as triangle detection. Therefore, the existence of any o(n(alpha))-time algorithm for this problem would be a significant breakthrough in the algorithmic field. Finally, our main result implies that planar graphs, bounded-treewidth graphs, and bounded-degree graphs can be decomposed by clique separators in linear or quasi-linear time.