Categories: Graph algorithms | NP-complete problems

Vertex cover problem

In computer science, the vertex cover problem is an NP-complete problem in complexity theory.

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

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

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.

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 $k$ and check each one to see whether it forms a vertex cover. 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 $k$ .

Vertex cover problem

See also

Further reading