Courcelle's theorem for triangulations

In graph theory, Courcelle's theorem essentially states that, if an algorithmic problem can be formulated in monadic second-order logic, then it can be solved in linear time for graphs of bounded treewidth. We prove such a metatheorem for a general class of triangulations of arbitrary fixed dim...

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Veröffentlicht in:Journal of combinatorial theory. Series A Jg. 146; S. 264 - 294
Hauptverfasser: Burton, Benjamin A., Downey, Rodney G.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.02.2017
Schlagworte:
3-Manifolds
Discrete Morse theory
Parameterised complexity
Triangulations
Turaev–Viro invariants
3-Manifolds
Turaev–Viro invariants
Discrete Morse theory
Triangulations
Parameterised complexity
ISSN:0097-3165, 1096-0899
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Zusammenfassung:In graph theory, Courcelle's theorem essentially states that, if an algorithmic problem can be formulated in monadic second-order logic, then it can be solved in linear time for graphs of bounded treewidth. We prove such a metatheorem for a general class of triangulations of arbitrary fixed dimension d, including all triangulated d-manifolds: if an algorithmic problem can be expressed in monadic second-order logic, then it can be solved in linear time for triangulations whose dual graphs have bounded treewidth. We apply our results to 3-manifold topology, a setting with many difficult computational problems but very few parameterised complexity results, and where treewidth has practical relevance as a parameter. Using our metatheorem, we recover and generalise earlier fixed-parameter tractability results on taut angle structures and discrete Morse theory respectively, and prove a new fixed-parameter tractability result for computing the powerful but complex Turaev–Viro invariants on 3-manifolds.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2016.10.001