We introduce an NP-complete special case of the Weighted Set Cover problem and show its fixed-parameter tractability with respect to the maximum subset size, a parameter that appears to be small in relevant applications. More precisely, in this practically relevant variant we require that the given collection C of subsets of a some base set S should be “tree-like.” That is, the subsets in C can...

In the classical vertex cover problem, we are given a graph G = (V,E) and we aim to find a minimum cardinality cover of the edges, i.e. a subset of the vertices C ⊆ V such that for every edge e ∈ E, at least one of its extremities belongs to C. In the Min-Power-Cover version of the vertex cover problem, we consider an edge-weighted graph and we aim to find a cover of the edges and a valuation (...

We describe a 3/2-approximation algorithm, LSE, for computing a b-Edge Cover of minimum weight in a graph with weights on the edges. The b-Edge Cover problem is a generalization of the better-known Edge Cover problem in graphs, where the objective is to choose a subset C of edges in the graph such that at least a specified number b(v) of edges in C are incident on each vertex v. In the weighted...

There are many polynomials which can be associated with a graph G, the most well known perhaps being the Tutte polynomial T (G; x, y) (cf. [B74] or [T54]). In particular, for specific values of x and y, T (G; x, y) enumerates various features of G. For example, T (G; 1, 1) is just the number of spanning trees of G, T (G; 2, 0) is the number of acyclic orientations of G, T (G; 1, 2) is the numbe...

A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a k-cycle cover k-DCC and k-UCC. Given a graph with edge weights one and two, Min-k-DCC and Min-k-UCC are the minimization problems of finding a k-cyc...

i∈F ci ≥ k for all F ∈ F(∆). If c is a (0, 1)-vector, then c may be identified with the subset C = {i ∈ [n] : ci 6= 0} of [n]. It is clear that c is a 1-cover if and only if C is a vertex cover of ∆ in the classical sense, that is, C ∩ F 6= ∅ for all F ∈ F(∆). Let S = K[x1, . . . , xn] be a polynomial ring in n variables over a field K. Let Ak(∆) denote the K-vector space generated by all monom...

It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining a polynomial kernel for Proper Interval Vertex Deletion. This resolves an open question of van Bevern, Komu...

Abstract Consider the following variant of the set cover problem. We are given a universe U = {1, ..., n} and a collection of subsets C = {S1, ..., Sm} where Si ⊆ U . For every element u ∈ U we need to find a set φ (u) ∈ C such that u ∈ φ (u). Once we construct and fix the mapping φ : U 7→ C a subset X ⊆ U of the universe is revealed, and we need to cover all elements from X with exactly φ(X) :...

Many known optimal NP-hardness of approximation results are reductions from a problem called LabelCover. The input is a bipartite graph G = (L,R,E) and each edge e = (x, y) ∈ E carries a projection πe that maps labels to x to labels to y. The objective is to find a labeling of the vertices that satisfies as many of the projections as possible. It is believed that the best approximation ratio ef...