Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/approximation/independent_set.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 r""" |
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2 Independent Set |
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3 |
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4 Independent set or stable set is a set of vertices in a graph, no two of |
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5 which are adjacent. That is, it is a set I of vertices such that for every |
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6 two vertices in I, there is no edge connecting the two. Equivalently, each |
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7 edge in the graph has at most one endpoint in I. The size of an independent |
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8 set is the number of vertices it contains. |
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9 |
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10 A maximum independent set is a largest independent set for a given graph G |
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11 and its size is denoted $\alpha(G)$. The problem of finding such a set is called |
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12 the maximum independent set problem and is an NP-hard optimization problem. |
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13 As such, it is unlikely that there exists an efficient algorithm for finding |
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14 a maximum independent set of a graph. |
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15 |
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16 `Wikipedia: Independent set <https://en.wikipedia.org/wiki/Independent_set_(graph_theory)>`_ |
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17 |
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18 Independent set algorithm is based on the following paper: |
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19 |
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20 $O(|V|/(log|V|)^2)$ apx of maximum clique/independent set. |
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21 |
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22 Boppana, R., & Halldórsson, M. M. (1992). |
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23 Approximating maximum independent sets by excluding subgraphs. |
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24 BIT Numerical Mathematics, 32(2), 180–196. Springer. |
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25 doi:10.1007/BF01994876 |
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26 |
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27 """ |
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28 from networkx.algorithms.approximation import clique_removal |
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29 |
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30 __all__ = ["maximum_independent_set"] |
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31 |
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32 |
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33 def maximum_independent_set(G): |
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34 """Returns an approximate maximum independent set. |
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35 |
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36 Parameters |
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37 ---------- |
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38 G : NetworkX graph |
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39 Undirected graph |
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40 |
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41 Returns |
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42 ------- |
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43 iset : Set |
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44 The apx-maximum independent set |
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45 |
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46 Notes |
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47 ----- |
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48 Finds the $O(|V|/(log|V|)^2)$ apx of independent set in the worst case. |
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49 |
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50 |
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51 References |
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52 ---------- |
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53 .. [1] Boppana, R., & Halldórsson, M. M. (1992). |
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54 Approximating maximum independent sets by excluding subgraphs. |
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55 BIT Numerical Mathematics, 32(2), 180–196. Springer. |
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56 """ |
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57 iset, _ = clique_removal(G) |
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58 return iset |