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The minimum vertex cover problem is the optimization problem of finding a smallest vertex cover in a ... an approximation algorithm with an approximation factor ...
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 ...
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. Covering problems allow the covering primitives to overlap; the process of covering something with non-overlapping primitives is called decomposition.
A vertex cover is a special case of a fractional vertex cover in which all weights are either 0 or 1. The size of a fractional vertex-cover is the sum of fractions of all vertices. The fractional vertex-cover number of a hypergraph H is the smallest size of a fractional vertex-cover in H. It is often denoted by τ*(H).
A w-vertex-cover is a multiset of vertices ("multiset" means that each vertex may appear several times), in which each edge e is adjacent to at least w e vertices. Egerváry's theorem says: In any edge-weighted bipartite graph, the maximum w-weight of a matching equals the smallest number of vertices in a w-vertex-cover.
Vertex covers can be represented by a bit string where each bit denotes whether the corresponding vertex is present in the cover. 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.
The problem is solved by the Hopcroft-Karp algorithm in time O(√ V E) time, and there are more efficient randomized algorithms, approximation algorithms, and algorithms for special classes of graphs such as bipartite planar graphs, as described in the main article.
For dense instances, however, there exists a -approximation algorithm for every >. [8] Tight example for the greedy algorithm with k=3. There is a standard example on which the greedy algorithm achieves an approximation ratio of /.