In computer science, the vertex cover problem or node cover problem is an NP-complete problem in complexity theory, and was one of Karp's 21 NP-complete problems.

A vertex cover of an undirected graph $G\; =\; (V,\; E)$ is a subset $V\text{'}$ of the vertices of the graph which contains at least one of the two endpoints of each edge:

$V\text{'}\; \backslash subseteq\; V:\; \backslash forall\; \backslash \{a,\; b\backslash \}\; \backslash in\; E\; :\; a\; \backslash in\; V\text{'}\; \backslash or\; b\; \backslash in\; V\text{'}$.

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 $G$ and a positive integer $k$.

QUESTION: Is there a vertex cover of size $k$ or less for $G$?

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 or, as Karp did, by reduction from the clique problem. As shown by Garey and Johnson in 1974, vertex cover remains NP-complete even in cubic graphs and even in planar graphs of degree at most 6.

Vertex cover is closely related to Independent Set problem by this theorem: a graph with $n$ vertices has a vertex cover of size $k$ if and only if the graph has an independent set of size $n-k$.

One can find a factor-2 approximation by repeatedly taking both endpoints of an edge into the vertex cover, then 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 choose some vertex and recursively branch into two cases: either take this vertex into the vertex cover, or all its neighbors. This algorithm is exponential in $k$, but not in the size of the graph, i.e., vertex cover is fixed-parameter tractable with respect to $k$.

See also

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• covering (graph theory)

References

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• 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.
• M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete problems. Proceedings of the sixth annual ACM symposium on Theory of computing, p.47-63. 1974.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 35.1, pp.1024–1027.
• A1.1: GT1, pg.190.

Graph algorithms | NP-complete problems

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