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• W2794549149 abstract "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 ${cal O}(nm)$-time and ${cal O}(n^{(3+alpha)/2}) = o(n^{2.69})$-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 ${cal O}(T(G) + min{n^{alpha},omega^2 n})$-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 ${cal O}(n^{alpha}log n)$-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." @default.
• W2794549149 created "2018-04-06" @default.
• W2794549149 creator A5020003818 @default.
• W2794549149 creator A5020955386 @default.
• W2794549149 date "2018-01-01" @default.
• W2794549149 modified "2023-10-18" @default.
• W2794549149 title "Revisiting Decomposition by Clique Separators" @default.
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• W2794549149 doi "https://doi.org/10.1137/16m1059837" @default.
• W2794549149 hasPublicationYear "2018" @default.
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