In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimization problem. It is NP-hard, so it cannot be solved by a polynomial-time algorithm if P ≠ NP. Moreover, it is hard to approximate – it cannot be approximated up to a factor smaller than 2 if the unique games conjecture is true. On the other hand, it has several simple 2-factor approximations. It is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory. The minimum vertex cover problem can be formulated as a half-integral, linear program whose dual linear program is the maximum matching problem. Vertex cover problems have been generalized to hypergraphs, see Vertex cover in hypergraphs.
Definition
Formally, a vertex cover V ′ {\displaystyle V'} of an undirected graph G = ( V , E ) {\displaystyle G=(V,E)} is a subset of V {\displaystyle V} such that ( u v ∈ E ) ⇒ ( u ∈ V ′ ∨ v ∈ V ′ ) {\displaystyle (uv\in E)\Rightarrow (u\in V'\lor v\in V')} , that is to say it is a set of vertices V ′ {\displaystyle V'} where every edge has at least one endpoint in the vertex cover V ′ {\displaystyle V'} . Such a set is said to cover the edges of G {\displaystyle G} . The upper figure shows two examples of vertex covers, with some vertex cover V ′ {\displaystyle V'} marked in red. A minimum vertex cover is a vertex cover of smallest possible size. The vertex cover number τ {\displaystyle \tau } is the size of a minimum vertex cover, i.e. τ = | V ′ | {\displaystyle \tau =|V'|} . The lower figure shows examples of minimum vertex covers in the previous graphs.
Examples The set of all vertices is a vertex cover. The endpoints of any maximal matching form a vertex cover. The complete bipartite graph K m , n {\displaystyle K_{m,n}} has a minimum vertex cover of size τ ( K m , n ) = min { m , n } {\displaystyle \tau (K_{m,n})=\min\{\,m,n\,\}} .
Properties A set of vertices is a vertex cover if and only if its complement is an independent set. Consequently, the number of vertices of a graph is equal to its minimum vertex cover number plus the size of a maximum independent set.
Computational problem The minimum vertex cover problem is the optimization problem of finding a smallest vertex cover in a given graph.
INSTANCE: Graph G {\displaystyle G}
OUTPUT: Smallest number k {\displaystyle k} such that G {\displaystyle G} has a vertex cover of size k {\displaystyle k} . If the problem is stated as a decision problem, it is called the vertex cover problem:
INSTANCE: Graph G {\displaystyle G} and positive integer k {\displaystyle k} . QUESTION: Does G {\displaystyle G} have a vertex cover of size at most k {\displaystyle k} ? They are equivalent under polynomial-time reduction by using binary search. The vertex cover problem is an NP-complete problem: it was one of Karp's 21 NP-complete problems. It is often used in computational complexity theory as a starting point for NP-hardness proofs.
ILP formulation Assume that every vertex has an associated cost of c ( v ) ≥ 0 {\displaystyle c(v)\geq 0} . The (weighted) minimum vertex cover problem can be formulated as the following integer linear program (ILP).
This ILP belongs to the more general class of ILPs for covering problems. The integrality gap of this ILP is 2 {\displaystyle 2} , so its relaxation (allowing each variable to be in the interval from 0 to 1, rather than requiring the variables to be only 0 or 1) gives a factor- 2 {\displaystyle 2} approximation algorithm for the minimum vertex cover problem. Furthermore, the linear programming relaxation of that ILP is half-integral, that is, there exists an optimal solution for which each entry x v {\displaystyle x_{v}} is either 0, 1/2, or 1. A 2-approximate vertex cover can be obtained from this fractional solution by selecting the subset of vertices whose variables are nonzero.
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