An edge cover of a graph is a set of edges in which each vertex has at least one of its incident edges. The problem of counting the number of edge covers is #P-complete and was shown to admit a fully polynomial-time approximation scheme (FPTAS) recently [10]. Counting weighted edge covers is the problem of computing the sum of the weights for all the edge covers, where the weight of each edge c...

Partial constraint satisfaction [5] was widely studied in the 90’s, and notably Max-CSP solving algorithms [21, 20, 1, 10]. These algorithms compute a lower bound of violated constraints without using propagation. Therefore, recent methods focus on the exploitation of propagation mechanisms to improve the solving process. Soft arc-consistency algorithms [11, 18, 19] propagate inconsistency coun...

We consider the minimum vertex cover problem in hypergraphs in which every hyperedge has size k (also known as minimum hitting set problem, or minimum set cover with element frequency k). Simple algorithms exist that provide k-approximations, and this is believed to be the best possible approximation achievable in polynomial time. We show how to exploit density and regularity properties of the ...

A graph is d-degenerate if its every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for DOMINATING SET in d-degenerate graphs, we investigate kernelization hardness of problems that include connectivity requirement in this class of graphs. Our m...

Let G be a graph. It is well known that the maximum multiplicity of a root of the matching polynomial μ(G,x) is at most the minimum number of vertex disjoint paths needed to cover the vertex set of G. Recently, a necessary and sufficient condition for which this bound is tight was found for trees. In this paper, a similar structural characterization is proved for any graph. To accomplish this, ...

The tree and tour cover problems on an edge-weighted graph are to compute a minimum weight tree and closed walk, respectively, whose vertices form a vertex cover. Both problems are NP-hard. In this note we give strongly polynomial time, constant factor approximation algorithms for both problems. An interesting feature of our algorithms is how they combine approximations of other problems, namel...

We study dense instances of several covering problems An in stance of the set cover problem with m sets is dense if there is such that any element of the ground set X belongs to at least m sets We show that the dense set cover problem can be approximated with the performance ratio c log jX j for any c though it is unlikely to be NP hard A polynomial time approximation scheme is constructed for ...

To determine if a graph has a spanning tree with few leaves is NP-hard. In this paper we study the parametric dual of this problem, k-INTERNAL SPANNING TREE (Does G have a spanning tree with at least k internal vertices?). We give an algorithm running in time O(2 log k · k + k · n). We also give a 2-approximation algorithm for the problem. However, the main contribution of this paper is that we...

The input of the Test Cover problem consists of a set V of vertices, and a collection E = {E1, . . . , Em} of distinct subsets of V , called tests. A test Eq separates a pair vi, vj of vertices if |{vi, vj} ∩ Eq| = 1. A subcollection T ⊆ E is a test cover if each pair vi, vj of distinct vertices is separated by a test in T . The objective is to find a test cover of minimum cardinality, if one e...

We describe an O(n log n) algorithm for the computation of the vertex separation of unicyclic graphs. The algorithm also computes a linear layout with optimal vertex separation in the same time bound. Pathwidth, node search number and vertex separation are different ways of defining the same notion. Path decompositions and search strategies can be derived from linear layouts. The algorithm appl...