Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulation problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques, and an integer t, it is possible in time the number of potential maximal cliques times $O(n^{O(t)})$ to find a maximum induced subgraph of treewidth t in G and for a given graph F of treewidth t, to decide if G contains an induced subgraph isomorphic to F. Combined with an improved algorithm enumerating all potential maximal cliques in time $O(1.734601^n)$, this yields that both the problems are solvable in time $1.734601^n$ * $n^{O(t)}$.
@InProceedings{fomin_et_al:LIPIcs.STACS.2010.2470, author = {Fomin, Fedor V. and Villanger, Yngve}, title = {{Finding Induced Subgraphs via Minimal Triangulations}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {383--394}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2470}, URN = {urn:nbn:de:0030-drops-24708}, doi = {10.4230/LIPIcs.STACS.2010.2470}, annote = {Keywords: Bounded treewidth, minimal triangulation, moderately exponential time algorithms} }
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