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. Formally, a vertex cover of an undirected graph is a subset of such that , that is to say it is a set of vertices where every edge has at least one endpoint in the vertex cover . Such a set is said to cover the edges of . The upper figure shows two examples of vertex covers, with some vertex cover marked in red. A minimum vertex cover is a vertex cover of smallest possible size. The vertex cover number is the size of a minimum vertex cover, i.e. . The lower figure shows examples of minimum vertex covers in the previous graphs. The set of all vertices is a vertex cover. The endpoints of any maximal matching form a vertex cover. The complete bipartite graph has a minimum vertex cover of size . 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 (Gallai 1959).