Results 1 
2 of
2
CONNECTED VERTEX COVERS IN DENSE GRAPHS
"... Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parame ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Abstract. We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorithm from Karpinski and Zelikovsky to the connected case. The new algorithm approximates the minimum connected vertex cover problem within a factor strictly less than 2 on all dense graphs. All these results are shown to be tight. Finally, we introduce the price of connectivity for the vertex cover problem, defined as the worstcase ratio between the sizes of a minimum connected vertex cover and a minimum vertex cover. We prove that the price of connectivity is bounded by 2/(1 + ε) in graphs with average degree εn, and give a family of neartight examples. Key words: approximation algorithm, vertex cover, connected vertex cover, dense graph. 1.
A tight analysis of the maximal matching heuristic
 IN PROC. OF THE ELEVENTH INTERNATIONAL COMPUTING AND COMBINATORICS CONFERENCE (COCOON), LNCS
, 2005
"... We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a wellknown 2approximation to three classical NPhard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. W ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
We study the algorithm that iteratively removes adjacent vertices from a simple, undirected graph until no edge remains. This algorithm is a wellknown 2approximation to three classical NPhard optimization problems: MINIMUM VERTEX COVER, MINIMUM MAXIMAL MATCHING and MINIMUM EDGE DOMINATING SET. We show that the worstcase approximation factor of this simple method can be expressed in a finer way when assumptions on the density of the graph is made. For graphs with an average degree at least ɛn, called weakly ɛdense graphs, we show that the asymptotic approximation factor is min{2, 1/(1 − √ 1 − ɛ)}. For graphs with a minimum degree at least ɛn – strongly ɛdense graphs – we show that the asymptotic approximation factor is min{2, 1/ɛ}. These bounds are obtained through a careful analysis of the tight examples.