We study the fixed-parameter tractability of the constraint satisfaction problem. We restrict constraint relations to languages in a family of NP-hard languages, classified by a purely combinatorial criterion that generalizes Boolean matrices with fixed row and column sum. For various natural and established structural parameterizations of the instances, we characterize the fixed-parameter trac...

This paper studies the fixed-parameter complexity of various problems in AI and nonmonotonic reasoning. We show that a number of relevant parameterized problems in these areas are fixedparameter tractable. Among these problems are constraint satisfaction problems with bounded treewidth and fixed domain, restricted satisfiability problems, propositional logic programming under the stable model s...

We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) structural restrictions on the interaction between the variables and the constraints and (ii) language restrictions on the relations that can be ...

Motivated by the prevalence of multi-layer network structures in biological and social systems, we investigate the problem of counting the number of occurrences of (small) subgraphs or motifs in multi-layer graphs in which each layer of the graph has useful structural properties. Making use of existing meta-theorems, we focus on the parameterised complexity of motif-counting problems, giving co...

Constraint satisfaction problems may be nearly tractable. For instance, most of the relations in a problem might belong to a tractable language. We introduce a method to take advantage of this fact by computing a backdoor to this tractable language. The method can be applied to many tractable classes for which the membership test is itself tractable. We introduce therefore two polynomial member...

We study the parameterized complexity of detecting backdoor sets for instances of the propositional satisfiability problem (SAT) with respect to the polynomially solvable classes horn and 2-cnf. A backdoor set is a subset of variables; for a strong backdoor set, the simplified formulas resulting from any setting of these variables is in a polynomially solvable class, and for a weak backdoor set...

In the present paper, we introduce the backdoor set approach into the field of temporal logic for the global fragment of linear temporal logic. We study the parameterized complexity of the satisfiability problem parameterized by the size of the backdoor. We distinguish between backdoor detection and evaluation of backdoors into the fragments of Horn and Krom formulas. Here we classify the opera...

In the Integer Quadratic Programming problem input is an n× n integer matrix Q, an m × n integer matrix A and an m-dimensional integer vector b. The task is to find a vector x ∈ Z minimizing xQx, subject to Ax ≤ b. We give a fixed parameter tractable algorithm for Integer Quadratic Programming parameterized by n+α. Here α is the largest absolute value of an entry of Q and A. As an application o...

We study the problem of finding a Pareto-efficient and envy-free allocation of a set of indivisible resources to a set of agents with monotonic preferences, either dichotomous or additive. Motivated by results of Bouveret and Lang [JAIR 2008], we provide a refined computational complexity analysis by studying the influence of three natural parameters: the number n of agents, the number m of res...

We consider the parameterized problems of whether a given set of clauses can be refuted within k resolution steps, and whether a given set of clauses contains an unsatisfiable subset of size at most k. We show that both problems are complete for the class W[1], the first level of the W-hierarchy of fixed-parameter intractable problems. Our results remain true if restricted to 3-SAT formulas and...