Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/approximation/vertex_cover.py @ 0:4f3585e2f14b draft default tip
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author  shellac 

date  Mon, 22 Mar 2021 18:12:50 +0000 
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1 """Functions for computing an approximate minimum weight vertex cover. 
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2 
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3 A vertex cover_ is a subset of nodes such that each edge in the graph 
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4 is incident to at least one node in the subset. 
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5 
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6 .. _vertex cover: https://en.wikipedia.org/wiki/Vertex_cover 
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7 .. vertex cover replace:: *vertex cover* 
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8 
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9 """ 
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10 
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11 __all__ = ["min_weighted_vertex_cover"] 
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12 
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13 
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14 def min_weighted_vertex_cover(G, weight=None): 
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15 r"""Returns an approximate minimum weighted vertex cover. 
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16 
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17 The set of nodes returned by this function is guaranteed to be a 
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18 vertex cover, and the total weight of the set is guaranteed to be at 
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19 most twice the total weight of the minimum weight vertex cover. In 
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20 other words, 
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21 
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22 .. math:: 
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23 
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24 w(S) \leq 2 * w(S^*), 
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25 
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26 where $S$ is the vertex cover returned by this function, 
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27 $S^*$ is the vertex cover of minimum weight out of all vertex 
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28 covers of the graph, and $w$ is the function that computes the 
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29 sum of the weights of each node in that given set. 
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30 
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31 Parameters 
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32  
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33 G : NetworkX graph 
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34 
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35 weight : string, optional (default = None) 
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36 If None, every node has weight 1. If a string, use this node 
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37 attribute as the node weight. A node without this attribute is 
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38 assumed to have weight 1. 
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39 
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40 Returns 
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41  
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42 min_weighted_cover : set 
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43 Returns a set of nodes whose weight sum is no more than twice 
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44 the weight sum of the minimum weight vertex cover. 
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45 
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46 Notes 
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47  
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48 For a directed graph, a vertex cover has the same definition: a set 
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49 of nodes such that each edge in the graph is incident to at least 
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50 one node in the set. Whether the node is the head or tail of the 
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51 directed edge is ignored. 
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52 
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53 This is the localratio algorithm for computing an approximate 
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54 vertex cover. The algorithm greedily reduces the costs over edges, 
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55 iteratively building a cover. The worstcase runtime of this 
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56 implementation is $O(m \log n)$, where $n$ is the number 
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57 of nodes and $m$ the number of edges in the graph. 
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58 
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59 References 
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60  
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61 .. [1] BarYehuda, R., and Even, S. (1985). "A localratio theorem for 
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62 approximating the weighted vertex cover problem." 
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63 *Annals of Discrete Mathematics*, 25, 27–46 
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64 <http://www.cs.technion.ac.il/~reuven/PDF/vc_lr.pdf> 
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65 
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66 """ 
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67 cost = dict(G.nodes(data=weight, default=1)) 
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68 # While there are uncovered edges, choose an uncovered and update 
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69 # the cost of the remaining edges. 
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70 for u, v in G.edges(): 
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71 min_cost = min(cost[u], cost[v]) 
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72 cost[u] = min_cost 
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73 cost[v] = min_cost 
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74 return {u for u, c in cost.items() if c == 0} 