constraint bipartite vertex cover is a graph-theoretical formalization of the spare allocation problem for reconfigurable arrays. We report on an implementation of a parameterized algorithm for this problem. This has led to considerable simplifications of the published, quite sophisticated algorithm. Moreover, we can prove that the mentioned algorithm could be quite efficient in practial situat...

A Minimal Cover by Triples is an ordered pair (V, T ) where V is a finite set and T is a collection of three-element subsets of V with the properties that every pair of elements of V appear together in at least one element of T and if any element of T is removed, the first property no longer holds. In this paper we explore the possible values |T | can take on for a given |V |. In addition, cons...

The calculation of an exact minimal cover of a Boolean function is an NP -complete problem. In this paper we introduce the definition of this problem and its basic solution. By using a slightly modified algorithm, we get a speed-up factor of more than 10. The main contributions of this paper are the description of an alternative approach mentioned in [15], and a remarkable improvement of this a...

We take a new look at the multicut problem in trees through the eyes of the vertex cover problem. This connection, together with other techniques that we develop, allows us to significantly improve the O(k) upper bound on the kernel size for multicut, given by Bousquet et al., to O(k). We exploit this connection further to present a parameterized algorithm for multicut that runs in time O∗(ρk),...

Let C be a clutter and let I be its edge ideal. We present a combinatorial description of the minimal generators of the symbolic Rees algebra Rs(I) of I . It is shown that the minimal generators of Rs(I) are in one to one correspondence with the irreducible parallelizations of C. From our description some major results on symbolic Rees algebras of perfect graphs and clutters will follow. As a b...

We study a probabilistic optimization model for graph-problems under vertex-uncertainty. We assume that any vertex vi of the input-graph G(V,E) has only a probability pi to be present in the final graph to be optimized (i.e., the final instance for the problem tackled will be only a sub-graph of the initial graph). Under this model, the original “deterministic” problem gives rise to a new (dete...

A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbors is in S. The problem of finding a defensive alliance of minimum size in a given graph is NP-hard and there are polynomial-time algorithms if certain parameters are bounded by a fixed constant. In particular, fixed-parameter tractability results have been obtained for some structural p...

The problem of finding a satisfying assignment for a 2-SAT formula that minimizes the number of variables that are set to 1 (min ones 2–sat) is NP-complete. It generalizes the well-studied problem of finding the smallest vertex cover of a graph, which can be modeled using a 2-SAT formula with no negative literals. The natural parameterized version of the problem asks for a satisfying assignment...

An edge dominating set of a graph G = (V,E) is a subset M ⊆ E of edges such that each edge in E \ M is incident to at least one edge in M . In this paper, we consider the parameterized edge dominating set problem which asks us to test whether a given graph has an edge dominating set with size bounded from above by an integer k or not, and we design an O(2.2351)-time and polynomial-space algorit...

We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The kPC problem is to find a k-fixed-endpoi...