Revisiting Decomposition by Clique Separators

Description

We study the complexity of decomposing a graph by means of clique separators. This common algorithmic tool, first introduced by Tarjan, allows 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+α)/2}) = o(n^2.69)-time complexity, with α < 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^α , ω^2 n})-time with T (G) and ω 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^α 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^α)-time algorithm for this problem would be a significant breakthrough in the field of algorithmic. 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.

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