Model counting for CNF formuals of bounded module treewidth.

The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity

Regular resolution for CNF of bounded incidence treewidth with few long clauses

We demonstrate that Regular Resolution is FPT for two restricted families of CNFs of bounded incidence treewidth. The first includes CNFs having at most $p$ clauses whose removal results in a CNF of primal treewidth at most $k$. The parameters we use in this case are $p$ and $k$. The second class includes CNFs of bounded one-sided (incidence) treewdth, a new parameter generalizing both primal treewidth and incidence pathwidth. The parameter we use in this case is the one-sided treewidth

Degree-3 Treewidth Sparsifiers

We study treewidth sparsifiers. Informally, given a graph $G$ of treewidth $k$, a treewidth sparsifier $H$ is a minor of $G$, whose treewidth is close to $k$, $|V(H)|$ is small, and the maximum vertex degree in $H$ is bounded. Treewidth sparsifiers of degree $3$ are of particular interest, as routing on node-disjoint paths, and computing minors seems easier in sub-cubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph $G$ of treewidth $k$, computes a topological minor $H$ of $G$ such that (i) the treewidth of $H$ is $\Omega(k/\text{polylog}(k))$; (ii) $|V(H)| = O(k^4)$; and (iii) the maximum vertex degree in $H$ is $3$. The running time of the algorithm is polynomial in $|V(G)|$ and $k$. Our result is in contrast to the known fact that unless $NP \subseteq coNP/{\sf poly}$, treewidth does not admit polynomial-size kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves node-disjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small good-quality vertex-cut sparsifiers that are also minors of the original graph.Comment: Extended abstract to appear in Proceedings of ACM-SIAM SODA 201