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

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...

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Published inDiscrete Applied Mathematics Vol. 160; no. 3; pp. 197 - 203
Main Authors Han, Qiaoming, Punnen, Abraham P.
Format Journal Article
LanguageEnglish
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
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Abstract 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.
AbstractList 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 [inline image], where sigma 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 sigma . We also obtain linear programming representations of the vertex cover problem on special graphs.
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.
Author Punnen, Abraham P.
Han, Qiaoming
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10.4086/toc.2006.v002a002
10.1016/j.orl.2009.01.010
10.1007/BF00290149
10.1137/S0097539700381097
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10.4007/annals.2005.162.439
10.1007/11427186_47
10.1145/502090.502098
10.1145/509907.510017
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Issue 3
Keywords 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
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Genetic linkage
NP complete problem
Combinatorics
Performance
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Snippet 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...
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...
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SubjectTerms 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
Title Strong and weak edges of a graph and linkages with the vertex cover problem
URI https://dx.doi.org/10.1016/j.dam.2011.10.027
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