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

Abstract

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is deÿnable 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 quantiÿcation. Such quantiÿcations are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this a ects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL deÿnable graph properties. Finally, our results are also applicable to SAT and ]SAT .

Model Counting for CNF Formulas of Bounded Modular Treewidth

“…We note that tractability of #SAT for formulas of bounded primal treewidth, bounded incidence treewidth, or bounded signed incidence clique-width can also be established using algorithmic meta-theorems by Courcelle, Makowsky, and Rotics [5,6]. The hardness result presented in Section 4 implies that our Theorem 1 cannot be proved in this way.…”

A Complete Characterisation of the Linear Clique-Width of Path Powers

“…Both clique-width and linear clique-width are NP-hard to compute [11]. These two closely related graph parameters have received much attention recently, and the interest in them is increasing [4,7,9,13,8,1,10,23,24,2,5,16,3,11,14,15,22,20,17,12].…”

Computing Graph Polynomials on Graphs of Bounded Clique-Width

“…Clique-width is a parameter of graphs similar, but more flexible, than its related notion of treewidth, [CO00], [CMR01]. Tree-width plays an important rôle in parametrized complexity theory, as shown forcefully in the monograph of Downey and Fellows [DF99].…”