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Example graph that has a vertex cover comprising 2 vertices (bottom), but none with fewer. 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.
For example, the bit string 0101 represents a cover consisting of the second and fourth vertex in a graph with four vertices. Sample graph to illustrate the minimum vertex cover problem. Consider the graph given in the figure. It has four vertices and there are two minimum vertex cover for this graph: vertices 0 and 2, and the vertices 1 and 2.
A simple example of an approximation algorithm is one for the minimum vertex cover problem, where the goal is to choose the smallest set of vertices such that every edge in the input graph contains at least one chosen vertex. One way to find a vertex cover is to repeat the following process: find an uncovered edge, add both its endpoints to the ...
Covering problems are minimization problems and usually integer linear programs, whose dual problems are called packing problems. The most prominent examples of covering problems are the set cover problem, which is equivalent to the hitting set problem, and its special cases, the vertex cover problem and the edge cover problem.
The complement of a vertex cover in any graph is an independent set, so a minimum vertex cover is complementary to a maximum independent set; finding maximum independent sets is another NP-complete problem. The equivalence between matching and covering articulated in Kőnig's theorem allows minimum vertex covers and maximum independent sets to ...
Vertex cover is another example for which iterative compression can be employed. In the vertex cover problem, a graph G = (V,E) and a natural number k are taken as inputs and the algorithm must decide whether there exists a set X of k vertices such that every edge is incident to a vertex in X.
In the fractional set cover problem, it is allowed to select fractions of sets, rather than entire sets. A fractional set cover is an assignment of a fraction (a number in [0,1]) to each set in , such that for each element x in the universe, the sum of fractions of sets that contain x is at least 1. The goal is to find a fractional set cover in ...
In graph theory, an edge cover of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. In computer science , the minimum edge cover problem is the problem of finding an edge cover of minimum size.