# Approximations and Heuristics¶

Approximations of graph properties and Heuristic functions for optimization problems.

Warning

The approximation submodule is not imported in the top-level

`networkx`

.These functions can be imported with

`from networkx.algorithms import approximation`

.

## Connectivity¶

Fast approximation for node connectivity

`all_pairs_node_connectivity` (G[, nbunch, cutoff]) |
Compute node connectivity between all pairs of nodes. |

`local_node_connectivity` (G, source, target[, …]) |
Compute node connectivity between source and target. |

`node_connectivity` (G[, s, t]) |
Returns an approximation for node connectivity for a graph or digraph G. |

## K-components¶

Fast approximation for k-component structure

`k_components` (G[, min_density]) |
Returns the approximate k-component structure of a graph G. |

## Clique¶

Functions for computing large cliques.

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

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

`large_clique_size` (G) |
Find the size of a large clique in a graph. |

## Clustering¶

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

## Dominating Set¶

Functions for finding node and edge dominating sets.

A dominating set for an undirected graph *G* with vertex set *V*
and edge set *E* is a subset *D* of *V* such that every vertex not in
*D* is adjacent to at least one member of *D*. An edge dominating set
is a subset *F* of *E* such that every edge not in *F* is
incident to an endpoint of at least one edge in *F*.

`min_weighted_dominating_set` (G[, weight]) |
Returns a dominating set that approximates the minimum weight node dominating set. |

`min_edge_dominating_set` (G) |
Returns 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.

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) |
Returns an approximate maximum independent set. |

## Matching¶

### Graph 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.

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

## Steiner Tree¶

`metric_closure` (G[, weight]) |
Return the metric closure of a graph. |

`steiner_tree` (G, terminal_nodes[, weight]) |
Return an approximation to the minimum Steiner tree of a graph. |

## Treewidth¶

Functions for computing treewidth decomposition.

Treewidth of an undirected graph is a number associated with the graph. It can be defined as the size of the largest vertex set (bag) in a tree decomposition of the graph minus one.

The notions of treewidth and tree decomposition have gained their attractiveness partly because many graph and network problems that are intractable (e.g., NP-hard) on arbitrary graphs become efficiently solvable (e.g., with a linear time algorithm) when the treewidth of the input graphs is bounded by a constant [1] [2].

There are two different functions for computing a tree decomposition:
`treewidth_min_degree()`

and `treewidth_min_fill_in()`

.

[1] | Hans L. Bodlaender and Arie M. C. A. Koster. 2010. “Treewidth computations I.Upper bounds”. Inf. Comput. 208, 3 (March 2010),259-275. http://dx.doi.org/10.1016/j.ic.2009.03.008 |

[2] | Hand L. Bodlaender. “Discovering Treewidth”. Institute of Information and Computing Sciences, Utrecht University. Technical Report UU-CS-2005-018. http://www.cs.uu.nl |

[3] | K. Wang, Z. Lu, and J. Hicks Treewidth.
http://web.eecs.utk.edu/~cphillip/cs594_spring2015_projects/treewidth.pdf |

`treewidth_min_degree` (G) |
Returns a treewidth decomposition using the Minimum Degree heuristic. |

`treewidth_min_fill_in` (G) |
Returns a treewidth decomposition using the Minimum Fill-in heuristic. |

## Vertex Cover¶

Functions for computing an approximate minimum weight vertex cover.

A *vertex cover* is a subset of nodes such that each edge in the graph
is incident to at least one node in the subset.

`min_weighted_vertex_cover` (G[, weight]) |
Returns an approximate minimum weighted vertex cover. |