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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.
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 /.
Nevertheless, in polynomial time it is possible to find an approximation with a ratio of 5/4. That is, this approximation algorithm finds a clique cover whose number of cliques is no more than 5/4 times the optimum. [4] Baker's technique can be used to provide a polynomial-time approximation scheme for the problem on planar graphs. [7]
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).
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 ...
[15] In the set cover problem formed from a metric dimension problem, the elements to be covered are the () pairs of vertices to be distinguished, and the sets that can cover them are the sets of pairs that can be distinguished by a single chosen vertex. The approximation bound then follows by applying standard approximation algorithms for set ...