U ?h/ã@sBdZddlmZddlmZddgZedƒd d d„ƒZd d„ZdS) aFunctions 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*. .. _dominating set: https://en.wikipedia.org/wiki/Dominating_set .. _edge dominating set: https://en.wikipedia.org/wiki/Edge_dominating_set é)Únot_implemented_foré)Úmaximal_matchingÚmin_weighted_dominating_setÚmin_edge_dominating_setZdirectedNcsxtˆƒdkrtƒStƒ‰‡‡‡fdd„}tˆƒ}‡fdd„ˆDƒ}|rtt| ¡|d\}}ˆ |¡||=||8}qBˆS)a[Returns a dominating set that approximates the minimum weight node dominating set. Parameters ---------- G : NetworkX graph Undirected graph. weight : string The node attribute storing the weight of an node. If provided, the node attribute with this key must be a number for each node. If not provided, each node is assumed to have weight one. Returns ------- min_weight_dominating_set : set A set of nodes, the sum of whose weights is no more than `(\log w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each node in the graph and `w(V^*)` denotes the sum of the weights of each node in the minimum weight dominating set. Notes ----- This algorithm computes an approximate minimum weighted dominating set for the graph `G`. The returned solution has weight `(\log w(V)) w(V^*)`, where `w(V)` denotes the sum of the weights of each node in the graph and `w(V^*)` denotes the sum of the weights of each node in the minimum weight dominating set for the graph. This implementation of the algorithm runs in $O(m)$ time, where $m$ is the number of edges in the graph. References ---------- .. [1] Vazirani, Vijay V. *Approximation Algorithms*. Springer Science & Business Media, 2001. écs&|\}}ˆj| ˆd¡t|ˆƒS)z¼Returns the cost-effectiveness of greedily choosing the given node. `node_and_neighborhood` is a two-tuple comprising a node and its closed neighborhood. é)ZnodesÚgetÚlen)Znode_and_neighborhoodÚvZ neighborhood©ÚGZdom_setÚweight©úb/var/www/html/venv/lib/python3.8/site-packages/networkx/algorithms/approximation/dominating_set.pyÚ_costEsz*min_weighted_dominating_set.._costcs i|]}||htˆ|ƒB“qSr)Úset)Ú.0r ©r rrÚ Usz/min_weighted_dominating_set..)Úkey)r rÚminÚitemsÚadd)r rrZverticesZ neighborhoodsZdom_nodeZmin_setrr rrs*    cCs|s tdƒ‚t|ƒS)aãReturns minimum cardinality edge dominating set. Parameters ---------- G : NetworkX graph Undirected graph Returns ------- min_edge_dominating_set : set Returns a set of dominating edges whose size is no more than 2 * OPT. Notes ----- The algorithm computes an approximate solution to the edge dominating set problem. The result is no more than 2 * OPT in terms of size of the set. Runtime of the algorithm is $O(|E|)$. z"Expected non-empty NetworkX graph!)Ú ValueErrorrrrrrrfs)N)Ú__doc__ÚutilsrZmatchingrÚ__all__rrrrrrÚs    P