Using the framework of kernelization we study whether efficient preprocessing schemes for the Treewidth problem can give provable bounds on the size of the processed instances. Assuming the ANDdistillation conjecture to hold, the standard parameterization of Treewidth does not have a kernel of polynomial size and thus instances (G, k) of the decision problem of Treewidth cannot be efficiently r...

Submodularity (or concavity) is considered as an important property in the field of cooperative game theory. In this article, we characterize submodular minimum coloring games and submodular minimum vertex cover games. These characterizations immediately show that it can be decided in polynomial time that the minimum coloring game or the minimum vertex cover game on a given graph is submodular ...

This paper gives the first polynomial time approximation scheme for the connected vertex cover problem in unit disk graphs. © 2009 Elsevier B.V. All rights reserved.

Approximation algorithms for a class of planar graph problems, including planar independent set, planar vertex cover and planar dominating set, were intensively studied. The current upper bound on the running time of the polynomial time approximation schemes (PTAS) for these planar graph problems is of 2 n. Here we study the lower bound on the running time of the PTAS for these planar graph pro...

The question of the existence of a polynomial kernelization of the Vertex Cover Above LP problem has been a longstanding, notorious open problem in Parameterized Complexity. Five years ago, the breakthrough work by Kratsch and Wahlström on representative sets has finally answered this question in the affirmative [FOCS 2012]. In this paper, we present an alternative, algebraic compression of the...

A function π : V → {1, . . . , k} is a broadcast coloring of order k if π(u) = π(v) implies that the distance between u and v is more than π(u). The minimum order of a broadcast coloring is called the broadcast chromatic number of G, and is denoted χb(G). In this paper we introduce this coloring and study its properties. In particular, we explore the relationship with the vertex cover and chrom...

We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. 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 ...

We explore opportunities for parameterising constant factor approximation algorithms for vertex cover. We provide a simple algorithm that works on any approximation ratio of the form 2l+1 l+1 and has complexity that outperforms an algorithm by Bourgeois et al. derived from a sophisticated exact parameterised algorithm. In particular, for l = 1 (factor 1.5 approximation) our algorithm runs in ti...

The dissociation number of a graph G is the number of vertices in a maximum size induced subgraph of G with vertex degree at most 1. A k-path vertex cover of a graph G is a subset S of vertices of G such that every path of order k in G contains at least one vertex from S. The minimum 3-path vertex cover is a dual problem to the dissociation number. For this problem we present an exact algorithm...