Stefan Mengel

Structural tractability of counting of solutions to conjunctive queries

In this paper we explore the problem of counting solutions to conjunctive queries. We consider a parameter called the quantified star size of a formula p which measures how the free variables are spread in p. We show that for conjunctive queries that admit nice decomposition properties (such as being of bounded treewidth or generalized hypertree width) bounded quantified star size exactly characterizes the classes of queries for which counting the number of solutions is tractable. This also allows us to fully characterize the conjunctive queries for which counting the solutions is tractable in the case of bounded arity. To illustrate the applicability of our results, we also show that computing the quantified star size of a formula is possible in time nO(k) for queries of generalized hypertree width k. Furthermore, quantified star size is even fixed parameter tractable parameterized by some other width measures, while it is W[1]-hard for generalized hypertree width and thus unlikely to be fixed parameter tractable. We finally show how to compute an approximation of quantified star size in polynomial time where the approximation ratio depends on the width of the input.

Understanding Model Counting for beta-acyclic CNF-formulas

We extend the knowledge about so-called structural restrictions of

A Trichotomy in the Complexity of Counting Answers to Conjunctive Queries

\mathrm{\#SAT}

Monomials in Arithmetic Circuits: Complete Problems in the Counting Hierarchy

by giving a polynomial time algorithm for

On Compiling CNFs into Structured Deterministic DNNFs

\beta

The complexity of weighted counting for acyclic conjunctive queries

-acyclic

Monomials in arithmetic circuits: Complete problems in the counting hierarchy.

\mathrm{\#SAT}

Hypergraph Acyclicity and Propositional Model Counting

. In contrast to previous algorithms in the area, our algorithm does not proceed by dynamic programming but works along an elimination order, solving a weighted version of constraint satisfaction. Moreover, we give evidence that this deviation from more standard algorithm is not a coincidence, but that there is likely no dynamic programming algorithm of the usual style for

Characterizing arithmetic circuit classes by constraint satisfaction problems

\beta

The Arithmetic Complexity of Tensor Contraction

-acyclic