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# Approximation¶

## Clique¶

Cliques.

`max_clique` (G) |
Find the Maximum Clique |

`clique_removal` (G) |
Repeatedly remove cliques from the graph. |

## Clustering¶

`average_clustering` (G[, trials]) |
Estimates the average clustering coefficient of G. |

## Dominating Set¶

A dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is joined to at least one member of D by some edge. The domination number gamma(G) is the number of vertices in a smallest dominating set for G. Given a graph G = (V, E) find a minimum weight dominating set V’.

http://en.wikipedia.org/wiki/Dominating_set

An edge dominating set for a graph G = (V, E) is a subset D of E such that every edge not in D is adjacent to at least one edge in D.

http://en.wikipedia.org/wiki/Edge_dominating_set

`min_weighted_dominating_set` (G[, weight]) |
Return minimum weight vertex dominating set. |

`min_edge_dominating_set` (G) |
Return minimum cardinality edge dominating set. |

## Independent Set¶

Independent Set

Independent set or stable set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set I of vertices such that for every two vertices in I, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in I. The size of an independent set is the number of vertices it contains.

A maximum independent set is a largest independent set for a given graph G and its size is denoted α(G). The problem of finding such a set is called the maximum independent set problem and is an NP-hard optimization problem. As such, it is unlikely that there exists an efficient algorithm for finding a maximum independent set of a graph.

http://en.wikipedia.org/wiki/Independent_set_(graph_theory)

Independent set algorithm is based on the following paper:

\(O(|V|/(log|V|)^2)\) apx of maximum clique/independent set.

Boppana, R., & Halldórsson, M. M. (1992). Approximating maximum independent sets by excluding subgraphs. BIT Numerical Mathematics, 32(2), 180–196. Springer. doi:10.1007/BF01994876

`maximum_independent_set` (G) |
Return an approximate maximum independent set. |

## Matching¶

Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex.

http://en.wikipedia.org/wiki/Matching_(graph_theory)

`min_maximal_matching` (G) |
Returns the minimum maximal matching of G. |

## Ramsey¶

Ramsey numbers.

`ramsey_R2` (G) |
Approximately computes the Ramsey number \(R(2;s,t)\) for graph. |

## Vertex Cover¶

Given an undirected graph \(G = (V, E)\) and a function w assigning nonnegative weights to its vertices, find a minimum weight subset of V such that each edge in E is incident to at least one vertex in the subset.

http://en.wikipedia.org/wiki/Vertex_cover

`min_weighted_vertex_cover` (G[, weight]) |
2-OPT Local Ratio for Minimum Weighted Vertex Cover |