Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/clique.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Functions for finding and manipulating cliques. |
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2 |
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3 Finding the largest clique in a graph is NP-complete problem, so most of |
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4 these algorithms have an exponential running time; for more information, |
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5 see the Wikipedia article on the clique problem [1]_. |
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6 |
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7 .. [1] clique problem:: https://en.wikipedia.org/wiki/Clique_problem |
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8 |
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9 """ |
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10 from collections import deque |
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11 from itertools import chain |
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12 from itertools import combinations |
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13 from itertools import islice |
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14 import networkx as nx |
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15 from networkx.utils import not_implemented_for |
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16 |
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17 |
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18 __all__ = [ |
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19 "find_cliques", |
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20 "find_cliques_recursive", |
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21 "make_max_clique_graph", |
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22 "make_clique_bipartite", |
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23 "graph_clique_number", |
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24 "graph_number_of_cliques", |
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25 "node_clique_number", |
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26 "number_of_cliques", |
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27 "cliques_containing_node", |
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28 "enumerate_all_cliques", |
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29 "max_weight_clique", |
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30 ] |
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31 |
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32 |
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33 @not_implemented_for("directed") |
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34 def enumerate_all_cliques(G): |
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35 """Returns all cliques in an undirected graph. |
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36 |
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37 This function returns an iterator over cliques, each of which is a |
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38 list of nodes. The iteration is ordered by cardinality of the |
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39 cliques: first all cliques of size one, then all cliques of size |
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40 two, etc. |
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41 |
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42 Parameters |
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43 ---------- |
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44 G : NetworkX graph |
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45 An undirected graph. |
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46 |
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47 Returns |
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48 ------- |
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49 iterator |
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50 An iterator over cliques, each of which is a list of nodes in |
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51 `G`. The cliques are ordered according to size. |
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52 |
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53 Notes |
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54 ----- |
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55 To obtain a list of all cliques, use |
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56 `list(enumerate_all_cliques(G))`. However, be aware that in the |
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57 worst-case, the length of this list can be exponential in the number |
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58 of nodes in the graph (for example, when the graph is the complete |
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59 graph). This function avoids storing all cliques in memory by only |
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60 keeping current candidate node lists in memory during its search. |
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61 |
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62 The implementation is adapted from the algorithm by Zhang, et |
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63 al. (2005) [1]_ to output all cliques discovered. |
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64 |
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65 This algorithm ignores self-loops and parallel edges, since cliques |
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66 are not conventionally defined with such edges. |
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67 |
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68 References |
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69 ---------- |
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70 .. [1] Yun Zhang, Abu-Khzam, F.N., Baldwin, N.E., Chesler, E.J., |
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71 Langston, M.A., Samatova, N.F., |
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72 "Genome-Scale Computational Approaches to Memory-Intensive |
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73 Applications in Systems Biology". |
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74 *Supercomputing*, 2005. Proceedings of the ACM/IEEE SC 2005 |
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75 Conference, pp. 12, 12--18 Nov. 2005. |
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76 <https://doi.org/10.1109/SC.2005.29>. |
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77 |
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78 """ |
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79 index = {} |
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80 nbrs = {} |
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81 for u in G: |
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82 index[u] = len(index) |
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83 # Neighbors of u that appear after u in the iteration order of G. |
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84 nbrs[u] = {v for v in G[u] if v not in index} |
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85 |
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86 queue = deque(([u], sorted(nbrs[u], key=index.__getitem__)) for u in G) |
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87 # Loop invariants: |
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88 # 1. len(base) is nondecreasing. |
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89 # 2. (base + cnbrs) is sorted with respect to the iteration order of G. |
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90 # 3. cnbrs is a set of common neighbors of nodes in base. |
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91 while queue: |
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92 base, cnbrs = map(list, queue.popleft()) |
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93 yield base |
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94 for i, u in enumerate(cnbrs): |
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95 # Use generators to reduce memory consumption. |
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96 queue.append( |
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97 ( |
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98 chain(base, [u]), |
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99 filter(nbrs[u].__contains__, islice(cnbrs, i + 1, None)), |
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100 ) |
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101 ) |
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102 |
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103 |
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104 @not_implemented_for("directed") |
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105 def find_cliques(G): |
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106 """Returns all maximal cliques in an undirected graph. |
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107 |
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108 For each node *v*, a *maximal clique for v* is a largest complete |
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109 subgraph containing *v*. The largest maximal clique is sometimes |
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110 called the *maximum clique*. |
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111 |
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112 This function returns an iterator over cliques, each of which is a |
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113 list of nodes. It is an iterative implementation, so should not |
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114 suffer from recursion depth issues. |
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115 |
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116 Parameters |
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117 ---------- |
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118 G : NetworkX graph |
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119 An undirected graph. |
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120 |
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121 Returns |
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122 ------- |
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123 iterator |
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124 An iterator over maximal cliques, each of which is a list of |
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125 nodes in `G`. The order of cliques is arbitrary. |
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126 |
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127 See Also |
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128 -------- |
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129 find_cliques_recursive |
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130 A recursive version of the same algorithm. |
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131 |
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132 Notes |
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133 ----- |
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134 To obtain a list of all maximal cliques, use |
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135 `list(find_cliques(G))`. However, be aware that in the worst-case, |
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136 the length of this list can be exponential in the number of nodes in |
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137 the graph. This function avoids storing all cliques in memory by |
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138 only keeping current candidate node lists in memory during its search. |
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139 |
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140 This implementation is based on the algorithm published by Bron and |
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141 Kerbosch (1973) [1]_, as adapted by Tomita, Tanaka and Takahashi |
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142 (2006) [2]_ and discussed in Cazals and Karande (2008) [3]_. It |
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143 essentially unrolls the recursion used in the references to avoid |
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144 issues of recursion stack depth (for a recursive implementation, see |
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145 :func:`find_cliques_recursive`). |
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146 |
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147 This algorithm ignores self-loops and parallel edges, since cliques |
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148 are not conventionally defined with such edges. |
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149 |
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150 References |
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151 ---------- |
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152 .. [1] Bron, C. and Kerbosch, J. |
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153 "Algorithm 457: finding all cliques of an undirected graph". |
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154 *Communications of the ACM* 16, 9 (Sep. 1973), 575--577. |
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155 <http://portal.acm.org/citation.cfm?doid=362342.362367> |
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156 |
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157 .. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi, |
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158 "The worst-case time complexity for generating all maximal |
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159 cliques and computational experiments", |
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160 *Theoretical Computer Science*, Volume 363, Issue 1, |
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161 Computing and Combinatorics, |
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162 10th Annual International Conference on |
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163 Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28--42 |
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164 <https://doi.org/10.1016/j.tcs.2006.06.015> |
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165 |
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166 .. [3] F. Cazals, C. Karande, |
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167 "A note on the problem of reporting maximal cliques", |
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168 *Theoretical Computer Science*, |
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169 Volume 407, Issues 1--3, 6 November 2008, Pages 564--568, |
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170 <https://doi.org/10.1016/j.tcs.2008.05.010> |
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171 |
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172 """ |
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173 if len(G) == 0: |
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174 return |
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175 |
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176 adj = {u: {v for v in G[u] if v != u} for u in G} |
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177 Q = [None] |
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178 |
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179 subg = set(G) |
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180 cand = set(G) |
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181 u = max(subg, key=lambda u: len(cand & adj[u])) |
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182 ext_u = cand - adj[u] |
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183 stack = [] |
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184 |
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185 try: |
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186 while True: |
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187 if ext_u: |
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188 q = ext_u.pop() |
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189 cand.remove(q) |
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190 Q[-1] = q |
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191 adj_q = adj[q] |
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192 subg_q = subg & adj_q |
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193 if not subg_q: |
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194 yield Q[:] |
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195 else: |
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196 cand_q = cand & adj_q |
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197 if cand_q: |
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198 stack.append((subg, cand, ext_u)) |
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199 Q.append(None) |
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200 subg = subg_q |
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201 cand = cand_q |
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202 u = max(subg, key=lambda u: len(cand & adj[u])) |
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203 ext_u = cand - adj[u] |
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204 else: |
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205 Q.pop() |
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206 subg, cand, ext_u = stack.pop() |
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207 except IndexError: |
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208 pass |
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209 |
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210 |
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211 # TODO Should this also be not implemented for directed graphs? |
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212 def find_cliques_recursive(G): |
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213 """Returns all maximal cliques in a graph. |
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214 |
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215 For each node *v*, a *maximal clique for v* is a largest complete |
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216 subgraph containing *v*. The largest maximal clique is sometimes |
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217 called the *maximum clique*. |
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218 |
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219 This function returns an iterator over cliques, each of which is a |
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220 list of nodes. It is a recursive implementation, so may suffer from |
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221 recursion depth issues. |
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222 |
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223 Parameters |
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224 ---------- |
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225 G : NetworkX graph |
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226 |
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227 Returns |
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228 ------- |
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229 iterator |
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230 An iterator over maximal cliques, each of which is a list of |
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231 nodes in `G`. The order of cliques is arbitrary. |
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232 |
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233 See Also |
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234 -------- |
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235 find_cliques |
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236 An iterative version of the same algorithm. |
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237 |
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238 Notes |
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239 ----- |
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240 To obtain a list of all maximal cliques, use |
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241 `list(find_cliques_recursive(G))`. However, be aware that in the |
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242 worst-case, the length of this list can be exponential in the number |
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243 of nodes in the graph. This function avoids storing all cliques in memory |
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244 by only keeping current candidate node lists in memory during its search. |
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245 |
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246 This implementation is based on the algorithm published by Bron and |
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247 Kerbosch (1973) [1]_, as adapted by Tomita, Tanaka and Takahashi |
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248 (2006) [2]_ and discussed in Cazals and Karande (2008) [3]_. For a |
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249 non-recursive implementation, see :func:`find_cliques`. |
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250 |
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251 This algorithm ignores self-loops and parallel edges, since cliques |
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252 are not conventionally defined with such edges. |
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253 |
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254 References |
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255 ---------- |
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256 .. [1] Bron, C. and Kerbosch, J. |
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257 "Algorithm 457: finding all cliques of an undirected graph". |
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258 *Communications of the ACM* 16, 9 (Sep. 1973), 575--577. |
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259 <http://portal.acm.org/citation.cfm?doid=362342.362367> |
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260 |
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261 .. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi, |
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262 "The worst-case time complexity for generating all maximal |
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263 cliques and computational experiments", |
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264 *Theoretical Computer Science*, Volume 363, Issue 1, |
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265 Computing and Combinatorics, |
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266 10th Annual International Conference on |
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267 Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28--42 |
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268 <https://doi.org/10.1016/j.tcs.2006.06.015> |
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269 |
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270 .. [3] F. Cazals, C. Karande, |
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271 "A note on the problem of reporting maximal cliques", |
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272 *Theoretical Computer Science*, |
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273 Volume 407, Issues 1--3, 6 November 2008, Pages 564--568, |
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274 <https://doi.org/10.1016/j.tcs.2008.05.010> |
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275 |
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276 """ |
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277 if len(G) == 0: |
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278 return iter([]) |
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279 |
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280 adj = {u: {v for v in G[u] if v != u} for u in G} |
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281 Q = [] |
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282 |
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283 def expand(subg, cand): |
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284 u = max(subg, key=lambda u: len(cand & adj[u])) |
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285 for q in cand - adj[u]: |
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286 cand.remove(q) |
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287 Q.append(q) |
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288 adj_q = adj[q] |
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289 subg_q = subg & adj_q |
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290 if not subg_q: |
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291 yield Q[:] |
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292 else: |
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293 cand_q = cand & adj_q |
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294 if cand_q: |
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295 yield from expand(subg_q, cand_q) |
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296 Q.pop() |
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297 |
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298 return expand(set(G), set(G)) |
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299 |
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300 |
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301 def make_max_clique_graph(G, create_using=None): |
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302 """Returns the maximal clique graph of the given graph. |
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303 |
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304 The nodes of the maximal clique graph of `G` are the cliques of |
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305 `G` and an edge joins two cliques if the cliques are not disjoint. |
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306 |
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307 Parameters |
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308 ---------- |
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309 G : NetworkX graph |
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310 |
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311 create_using : NetworkX graph constructor, optional (default=nx.Graph) |
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312 Graph type to create. If graph instance, then cleared before populated. |
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313 |
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314 Returns |
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315 ------- |
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316 NetworkX graph |
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317 A graph whose nodes are the cliques of `G` and whose edges |
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318 join two cliques if they are not disjoint. |
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319 |
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320 Notes |
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321 ----- |
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322 This function behaves like the following code:: |
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323 |
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324 import networkx as nx |
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325 G = nx.make_clique_bipartite(G) |
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326 cliques = [v for v in G.nodes() if G.nodes[v]['bipartite'] == 0] |
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327 G = nx.bipartite.project(G, cliques) |
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328 G = nx.relabel_nodes(G, {-v: v - 1 for v in G}) |
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329 |
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330 It should be faster, though, since it skips all the intermediate |
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331 steps. |
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332 |
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333 """ |
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334 if create_using is None: |
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335 B = G.__class__() |
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336 else: |
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337 B = nx.empty_graph(0, create_using) |
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338 cliques = list(enumerate(set(c) for c in find_cliques(G))) |
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339 # Add a numbered node for each clique. |
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340 B.add_nodes_from(i for i, c in cliques) |
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341 # Join cliques by an edge if they share a node. |
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342 clique_pairs = combinations(cliques, 2) |
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343 B.add_edges_from((i, j) for (i, c1), (j, c2) in clique_pairs if c1 & c2) |
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344 return B |
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345 |
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346 |
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347 def make_clique_bipartite(G, fpos=None, create_using=None, name=None): |
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348 """Returns the bipartite clique graph corresponding to `G`. |
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349 |
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350 In the returned bipartite graph, the "bottom" nodes are the nodes of |
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351 `G` and the "top" nodes represent the maximal cliques of `G`. |
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352 There is an edge from node *v* to clique *C* in the returned graph |
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353 if and only if *v* is an element of *C*. |
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354 |
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355 Parameters |
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356 ---------- |
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357 G : NetworkX graph |
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358 An undirected graph. |
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359 |
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360 fpos : bool |
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361 If True or not None, the returned graph will have an |
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362 additional attribute, `pos`, a dictionary mapping node to |
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363 position in the Euclidean plane. |
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364 |
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365 create_using : NetworkX graph constructor, optional (default=nx.Graph) |
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366 Graph type to create. If graph instance, then cleared before populated. |
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367 |
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368 Returns |
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369 ------- |
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370 NetworkX graph |
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371 A bipartite graph whose "bottom" set is the nodes of the graph |
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372 `G`, whose "top" set is the cliques of `G`, and whose edges |
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373 join nodes of `G` to the cliques that contain them. |
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374 |
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375 The nodes of the graph `G` have the node attribute |
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376 'bipartite' set to 1 and the nodes representing cliques |
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377 have the node attribute 'bipartite' set to 0, as is the |
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378 convention for bipartite graphs in NetworkX. |
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379 |
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380 """ |
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381 B = nx.empty_graph(0, create_using) |
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382 B.clear() |
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383 # The "bottom" nodes in the bipartite graph are the nodes of the |
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384 # original graph, G. |
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385 B.add_nodes_from(G, bipartite=1) |
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386 for i, cl in enumerate(find_cliques(G)): |
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387 # The "top" nodes in the bipartite graph are the cliques. These |
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388 # nodes get negative numbers as labels. |
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389 name = -i - 1 |
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390 B.add_node(name, bipartite=0) |
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391 B.add_edges_from((v, name) for v in cl) |
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392 return B |
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393 |
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394 |
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395 def graph_clique_number(G, cliques=None): |
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396 """Returns the clique number of the graph. |
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397 |
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398 The *clique number* of a graph is the size of the largest clique in |
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399 the graph. |
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400 |
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401 Parameters |
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402 ---------- |
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403 G : NetworkX graph |
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404 An undirected graph. |
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405 |
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406 cliques : list |
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407 A list of cliques, each of which is itself a list of nodes. If |
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408 not specified, the list of all cliques will be computed, as by |
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409 :func:`find_cliques`. |
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410 |
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411 Returns |
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412 ------- |
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413 int |
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414 The size of the largest clique in `G`. |
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415 |
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416 Notes |
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417 ----- |
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418 You should provide `cliques` if you have already computed the list |
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419 of maximal cliques, in order to avoid an exponential time search for |
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420 maximal cliques. |
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421 |
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422 """ |
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423 if cliques is None: |
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424 cliques = find_cliques(G) |
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425 if len(G.nodes) < 1: |
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426 return 0 |
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427 return max([len(c) for c in cliques] or [1]) |
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428 |
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429 |
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430 def graph_number_of_cliques(G, cliques=None): |
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431 """Returns the number of maximal cliques in the graph. |
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432 |
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433 Parameters |
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434 ---------- |
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435 G : NetworkX graph |
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436 An undirected graph. |
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437 |
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438 cliques : list |
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439 A list of cliques, each of which is itself a list of nodes. If |
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440 not specified, the list of all cliques will be computed, as by |
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441 :func:`find_cliques`. |
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442 |
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443 Returns |
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444 ------- |
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445 int |
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446 The number of maximal cliques in `G`. |
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447 |
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448 Notes |
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449 ----- |
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450 You should provide `cliques` if you have already computed the list |
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451 of maximal cliques, in order to avoid an exponential time search for |
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452 maximal cliques. |
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453 |
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454 """ |
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455 if cliques is None: |
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456 cliques = list(find_cliques(G)) |
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457 return len(cliques) |
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458 |
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459 |
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460 def node_clique_number(G, nodes=None, cliques=None): |
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461 """ Returns the size of the largest maximal clique containing |
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462 each given node. |
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463 |
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464 Returns a single or list depending on input nodes. |
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465 Optional list of cliques can be input if already computed. |
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466 """ |
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467 if cliques is None: |
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468 if nodes is not None: |
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469 # Use ego_graph to decrease size of graph |
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470 if isinstance(nodes, list): |
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471 d = {} |
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472 for n in nodes: |
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473 H = nx.ego_graph(G, n) |
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474 d[n] = max(len(c) for c in find_cliques(H)) |
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475 else: |
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476 H = nx.ego_graph(G, nodes) |
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477 d = max(len(c) for c in find_cliques(H)) |
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478 return d |
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479 # nodes is None--find all cliques |
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480 cliques = list(find_cliques(G)) |
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481 |
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482 if nodes is None: |
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483 nodes = list(G.nodes()) # none, get entire graph |
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484 |
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485 if not isinstance(nodes, list): # check for a list |
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486 v = nodes |
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487 # assume it is a single value |
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488 d = max([len(c) for c in cliques if v in c]) |
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489 else: |
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490 d = {} |
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491 for v in nodes: |
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492 d[v] = max([len(c) for c in cliques if v in c]) |
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493 return d |
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494 |
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495 # if nodes is None: # none, use entire graph |
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496 # nodes=G.nodes() |
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497 # elif not isinstance(nodes, list): # check for a list |
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498 # nodes=[nodes] # assume it is a single value |
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499 |
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500 # if cliques is None: |
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501 # cliques=list(find_cliques(G)) |
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502 # d={} |
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503 # for v in nodes: |
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504 # d[v]=max([len(c) for c in cliques if v in c]) |
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505 |
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506 # if nodes in G: |
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507 # return d[v] #return single value |
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508 # return d |
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509 |
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510 |
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511 def number_of_cliques(G, nodes=None, cliques=None): |
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512 """Returns the number of maximal cliques for each node. |
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513 |
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514 Returns a single or list depending on input nodes. |
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515 Optional list of cliques can be input if already computed. |
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516 """ |
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517 if cliques is None: |
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518 cliques = list(find_cliques(G)) |
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519 |
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520 if nodes is None: |
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521 nodes = list(G.nodes()) # none, get entire graph |
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522 |
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523 if not isinstance(nodes, list): # check for a list |
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524 v = nodes |
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525 # assume it is a single value |
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526 numcliq = len([1 for c in cliques if v in c]) |
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527 else: |
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528 numcliq = {} |
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529 for v in nodes: |
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530 numcliq[v] = len([1 for c in cliques if v in c]) |
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531 return numcliq |
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532 |
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533 |
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534 def cliques_containing_node(G, nodes=None, cliques=None): |
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535 """Returns a list of cliques containing the given node. |
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536 |
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537 Returns a single list or list of lists depending on input nodes. |
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538 Optional list of cliques can be input if already computed. |
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539 """ |
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540 if cliques is None: |
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541 cliques = list(find_cliques(G)) |
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542 |
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543 if nodes is None: |
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544 nodes = list(G.nodes()) # none, get entire graph |
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545 |
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546 if not isinstance(nodes, list): # check for a list |
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547 v = nodes |
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548 # assume it is a single value |
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549 vcliques = [c for c in cliques if v in c] |
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550 else: |
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551 vcliques = {} |
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552 for v in nodes: |
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553 vcliques[v] = [c for c in cliques if v in c] |
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554 return vcliques |
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555 |
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556 |
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557 class MaxWeightClique(object): |
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558 """A class for the maximum weight clique algorithm. |
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559 |
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560 This class is a helper for the `max_weight_clique` function. The class |
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561 should not normally be used directly. |
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562 |
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563 Parameters |
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564 ---------- |
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565 G : NetworkX graph |
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566 The undirected graph for which a maximum weight clique is sought |
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567 weight : string or None, optional (default='weight') |
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568 The node attribute that holds the integer value used as a weight. |
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569 If None, then each node has weight 1. |
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570 |
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571 Attributes |
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572 ---------- |
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573 G : NetworkX graph |
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574 The undirected graph for which a maximum weight clique is sought |
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575 node_weights: dict |
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576 The weight of each node |
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577 incumbent_nodes : list |
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578 The nodes of the incumbent clique (the best clique found so far) |
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579 incumbent_weight: int |
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580 The weight of the incumbent clique |
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581 """ |
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582 |
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583 def __init__(self, G, weight): |
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584 self.G = G |
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585 self.incumbent_nodes = [] |
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586 self.incumbent_weight = 0 |
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587 |
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588 if weight is None: |
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589 self.node_weights = {v: 1 for v in G.nodes()} |
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590 else: |
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591 for v in G.nodes(): |
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592 if weight not in G.nodes[v]: |
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593 err = "Node {} does not have the requested weight field." |
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594 raise KeyError(err.format(v)) |
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595 if not isinstance(G.nodes[v][weight], int): |
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596 err = "The '{}' field of node {} is not an integer." |
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597 raise ValueError(err.format(weight, v)) |
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598 self.node_weights = {v: G.nodes[v][weight] for v in G.nodes()} |
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599 |
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600 def update_incumbent_if_improved(self, C, C_weight): |
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601 """Update the incumbent if the node set C has greater weight. |
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602 |
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603 C is assumed to be a clique. |
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604 """ |
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605 if C_weight > self.incumbent_weight: |
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606 self.incumbent_nodes = C[:] |
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607 self.incumbent_weight = C_weight |
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608 |
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609 def greedily_find_independent_set(self, P): |
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610 """Greedily find an independent set of nodes from a set of |
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611 nodes P.""" |
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612 independent_set = [] |
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613 P = P[:] |
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614 while P: |
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615 v = P[0] |
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616 independent_set.append(v) |
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617 P = [w for w in P if v != w and not self.G.has_edge(v, w)] |
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618 return independent_set |
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619 |
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620 def find_branching_nodes(self, P, target): |
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621 """Find a set of nodes to branch on.""" |
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622 residual_wt = {v: self.node_weights[v] for v in P} |
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623 total_wt = 0 |
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624 P = P[:] |
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625 while P: |
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626 independent_set = self.greedily_find_independent_set(P) |
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627 min_wt_in_class = min(residual_wt[v] for v in independent_set) |
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628 total_wt += min_wt_in_class |
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629 if total_wt > target: |
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630 break |
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631 for v in independent_set: |
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632 residual_wt[v] -= min_wt_in_class |
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633 P = [v for v in P if residual_wt[v] != 0] |
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634 return P |
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635 |
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636 def expand(self, C, C_weight, P): |
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637 """Look for the best clique that contains all the nodes in C and zero or |
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638 more of the nodes in P, backtracking if it can be shown that no such |
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639 clique has greater weight than the incumbent. |
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640 """ |
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641 self.update_incumbent_if_improved(C, C_weight) |
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642 branching_nodes = self.find_branching_nodes(P, self.incumbent_weight - C_weight) |
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643 while branching_nodes: |
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644 v = branching_nodes.pop() |
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645 P.remove(v) |
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646 new_C = C + [v] |
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647 new_C_weight = C_weight + self.node_weights[v] |
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648 new_P = [w for w in P if self.G.has_edge(v, w)] |
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649 self.expand(new_C, new_C_weight, new_P) |
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650 |
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651 def find_max_weight_clique(self): |
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652 """Find a maximum weight clique.""" |
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653 # Sort nodes in reverse order of degree for speed |
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654 nodes = sorted(self.G.nodes(), key=lambda v: self.G.degree(v), reverse=True) |
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655 nodes = [v for v in nodes if self.node_weights[v] > 0] |
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656 self.expand([], 0, nodes) |
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657 |
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658 |
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659 @not_implemented_for("directed") |
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660 def max_weight_clique(G, weight="weight"): |
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661 """Find a maximum weight clique in G. |
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662 |
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663 A *clique* in a graph is a set of nodes such that every two distinct nodes |
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664 are adjacent. The *weight* of a clique is the sum of the weights of its |
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665 nodes. A *maximum weight clique* of graph G is a clique C in G such that |
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666 no clique in G has weight greater than the weight of C. |
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667 |
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668 Parameters |
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669 ---------- |
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670 G : NetworkX graph |
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671 Undirected graph |
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672 weight : string or None, optional (default='weight') |
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673 The node attribute that holds the integer value used as a weight. |
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674 If None, then each node has weight 1. |
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675 |
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676 Returns |
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677 ------- |
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678 clique : list |
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679 the nodes of a maximum weight clique |
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680 weight : int |
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681 the weight of a maximum weight clique |
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682 |
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683 Notes |
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684 ----- |
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685 The implementation is recursive, and therefore it may run into recursion |
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686 depth issues if G contains a clique whose number of nodes is close to the |
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687 recursion depth limit. |
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688 |
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689 At each search node, the algorithm greedily constructs a weighted |
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690 independent set cover of part of the graph in order to find a small set of |
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691 nodes on which to branch. The algorithm is very similar to the algorithm |
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692 of Tavares et al. [1]_, other than the fact that the NetworkX version does |
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693 not use bitsets. This style of algorithm for maximum weight clique (and |
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694 maximum weight independent set, which is the same problem but on the |
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695 complement graph) has a decades-long history. See Algorithm B of Warren |
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696 and Hicks [2]_ and the references in that paper. |
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697 |
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698 References |
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699 ---------- |
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700 .. [1] Tavares, W.A., Neto, M.B.C., Rodrigues, C.D., Michelon, P.: Um |
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701 algoritmo de branch and bound para o problema da clique máxima |
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702 ponderada. Proceedings of XLVII SBPO 1 (2015). |
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703 |
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704 .. [2] Warrent, Jeffrey S, Hicks, Illya V.: Combinatorial Branch-and-Bound |
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705 for the Maximum Weight Independent Set Problem. Technical Report, |
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706 Texas A&M University (2016). |
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707 """ |
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708 |
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709 mwc = MaxWeightClique(G, weight) |
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710 mwc.find_max_weight_clique() |
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711 return mwc.incumbent_nodes, mwc.incumbent_weight |