We study the computational problem of checking whether a logical sentence is true in a finite abelian group. We prove that model checking first-order sentences on finite abelian groups is fixed-parameter tractable, when parameterized by the size of the sentence. We also prove that model checking monadic second-order sentences on finite abelian groups finitely presented by integer matrices is no...

We study the problems Locating-Dominating Set and Metric Dimension, which consist in determining a minimum-size set of vertices that distinguishes the vertices of a graph using either neighbourhoods or distances. We consider these problems when restricted to interval graphs and permutation graphs. We prove that both decision problems are NP-complete, even for graphs that are at the same time in...

Graph complexity measures like tree-width, clique-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. These algorithms are based on hierarchical decompositions of the considered graphs and on boundedness conditions on the graph operations that, more or less explicitly, recombine the components of decompositions into larger graphs. Rank-width is de ned in ...

In the Mixed Chinese Postman Problem (MCPP), given an edge-weighted mixed graph G (G may have both edges and arcs), our aim is to find a minimum weight closed walk traversing each edge and arc at least once. The MCPP parameterized by the number of edges was known to be fixed-parameter tractable using a simple argument. Solving an open question of van Bevern et al., we prove that the MCPP parame...

We consider the parameterized complexity of the Unique Coverage problem: given a family of sets and a parameter k, we ask whether there exists a subfamily that covers at least k elements exactly once. This NP-complete problem has applications in wireless networks and radio broadcasting and is also a natural generalization of the well-known Max Cut problem. We show that this problem is fixed-par...

In the problemMin-DESC, we are given a digraphD and an integer k, and asked whether there exists a set A of at most k arcs inD, such that if we remove the arcs of A, in the resulting digraph every strong component is Eulerian. Min-DESC is NP-hard; Cechlárová and Schlotter (IPEC 2010) asked whether the problem is fixed-parameter tractable when parameterized by k. We consider the subproblem of Mi...

In this paper, we introduce a notion of backdoors to Reiter’s propositional default logic and study structural properties of it. Also we consider the problems of backdoor detection (parameterised by the solution size) as well as backdoor evaluation (parameterised by the size of the given backdoor), for various kinds of target classes (cnf, horn, krom, monotone, positive-unit). We show that back...

We show that every problem in MAX SNP has a lower bound on the optimum solution size that is unbounded and that the above guarantee question with respect to this lower bound is fixed parameter tractable. We next introduce the notion of “tight” upper and lower bounds for the optimum solution and show that the parameterized version of a variant of the above guarantee question with respect to the ...

Graph polynomials which are definable in Monadic Second Order Logic (MSOL) on the vocabulary of graphs are Fixed-Parameter Tractable (FPT) with respect to clique-width. In contrast, graph polynomials which are definable in MSOL on the vocabulary of hypergraphs are fixed-parameter tractable with respect to tree-width, but not necessarily with respect to clique-width. No algorithmic meta-theorem ...

We construct a fixed parameter algorithm parameterized by d and k that takes as an input a graph G′ obtained from a d-degenerate graph G by complementing on at most k arbitrary subsets of the vertex set of G and outputs a graph H such that G and H agree on all but f(d, k) vertices. Our work is motivated by the first order model checking in graph classes that are first order interpretable in cla...