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Vertex Cover ProblemIn computer science, the vertex cover problem is an NP-complete problem in complexity theory. A vertex cover of an undirected graph is a subset of the vertices of the graph which contains at least one of the two endpoints of each edge: - .
In the graph at the right, {1,3,5,6} is an example of a vertex cover. The vertex cover problem is the optimization problem of finding a vertex cover of minimum size in a graph. The problem can also be stated as a decision problem: - INSTANCE: A graph and a positive integer .
- QUESTION: Is there a vertex cover of size or less for ?
Vertex cover is NP-complete, which means it is unlikely that there is an efficient algorithm to solve it. NP-completeness can be proven by reduction from 3-satisfiability. Vertex cover is closely related to Independent Set problem by the theorem - a graph with n vertices has a vertex cover of size k if and only if the graph has an independence set of size n-k. One can find a factor-2 approximation by repeatedly taking both endpoints of an edge into the vertex cover, removing them from the graph. No better constant-factor approximation is known; the problem is APX-complete, i.e., it cannot be approximated arbitrarily well. A brute force algorithm to find a vertex cover in a graph is to list all subsets of vertices of size and check each one to see whether it forms a vertex cover. This algorithm is exponential in , but not in the size of the graph, i.e., vertex cover is fixed-parameter tractable with respect to . See also Further reading - Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, 1979.
- Preeti Patil & Sangita Patil. (Lecturer in Vartak Polytechnic) A Guide to the Theory of NP-Completeness. BPB & Co., Mumbai, 2004.
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