On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is definable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 108; no. 1; pp. 23 - 52
Main Authors: Courcelle, B., Makowsky, J.A., Rotics, U.
Format: Journal Article
Language:English
Published: Elsevier B.V 15.02.2001
Subjects:
Combinatorial enumeration
Fixed parameter complexity
Combinatorial enumeration
Fixed parameter complexity
ISSN:0166-218X, 1872-6771
Online Access:Get full text
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Summary:We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is definable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quantification. Such quantifications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this affects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL definable graph properties. Finally, our results are also applicable to SAT and ♯ SAT.
ISSN:0166-218X
1872-6771
DOI:10.1016/S0166-218X(00)00221-3