Strong and weak edges of a graph and linkages with the vertex cover problem

• Published in: Discrete Applied Mathematics Vol. 160; no. 3; pp. 197 - 203
• Main Authors: Han, Qiaoming, Punnen, Abraham P.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.02.2012; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Approximation algorithm; Approximations and expansions; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graphs; LP-relaxation; Mathematical analysis; Mathematics; NP-complete problems; Optimization; Representations; Sciences and techniques of general use; Theoretical computing; Upper bounds; Vertex cover problem; Weak edge reduction; Weak edge reduction; NP-complete problems; LP-relaxation; Approximation algorithm; Vertex cover problem; Vertex; Edge(graph); Computer theory; Polynomial approximation; Linear programming; Vertex(graph); Optimization; Polynomial time; Relaxation; Upper bound; Genetic linkage; NP complete problem; Combinatorics; Performance
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2011.10.027

In this paper we show that the problem of identifying an edge ( i , j ) of a graph G such that there exists an optimal vertex cover S of G containing exactly one of the vertices i and j is NP-hard. Such an edge is called a weak edge. We then develop a polynomial time approximation algorithm for the vertex cover problem with performance guarantee 2 − 1 1 + σ , where σ is an upper bound on a measure related to a weak edge of a graph. A related problem of identifying an edge ( i , j ) such that there exists an optimal vertex cover containing both vertices i and j is also shown to be NP-hard. Further, we discuss a new relaxation of the vertex cover problem which is used in our approximation algorithm to obtain smaller values of σ . We also obtain linear programming representations of the vertex cover problem on special graphs.