This repository contains functions used to approximate the minimum feedback vertex set (FVS) of a directed graph, a problem whose exact solution is known to be NP-hard. This algorithms was used to estimate the size of the minimum FVS in a recent paper "Zañudo, J. G. T., Yang, G., & Albert, R. (2017). PNAS 114: 28, 7234–7239, doi: 10.1073/pnas.1617387114".
This algorithm identifies a near-minimum feedback vertex set using simulated annealing (SA) and a local search of topological ordering. The algorithm is describe in the paper "Galinier, P., Lemamou, E. & Bouzidi, M.W. J Heuristics (2013) 19: 797. doi:10.1007/s10732-013-9224-z". The code follows the pseudocode given in Page 805 in that paper.
The code is written in Python 2.7. The module requires NetworkX 1.11.
Another version written in Cython is available at https://github.com/yanggangthu/FVS_cython. The cython version is typically 5-10 times faster than the python version.
Related functions are stored in FVS.py and FVS_localsearch_10_python.py.
Core functions of finding a maximum sub topological ordering of a given graph (which is equivalent to identifying a minimum FVS) is written in FVS_localsearch_10_python.py.
FVS.py is a wrapper, that deals with graph input and identifies a near-minimum FVS by subtracting the nodes in the topological ordering from all the nodes.
FVS_test.py contains three examples illustrating how to use the code.
To use this module, import FVS and call FVS() just as a regular function.
The function can take 6 paramters listed below and only the first (the graph) is neccessary.
G : NetworkX Graph/DiGraph, result for MultiGraph is not tested
T_0 : the initial temperature in SA
alpha : the cooling multiplier for temperatue in the geometric cooling regime
maxMvt_factor : maxMvt_factor times network size is the number of iterations for the inner loop given a fixed temperatue
maxFail : FVS_local_search stops when maxFail number of outloops (temperatue) fail to improve the result
randomseed: random seed for generating random numbers
T_0 = 0.6, alpha = 0.99, maxMvt_factor = 5, maxFail = 50, randomseed=None
The default values are suggested by the author of the paper.
T_0 and maxFail are chosen after a limited number of preliminary experiments
alpha is chosen more arbitrarily, however alpha should be a positive number slightly smaller than 1.
Increase alpha or maxMvt_factor or maxFail will increase the time of finding FVS.
An approximation of the minimum FVS of the given graph as a list.
import networkx as nx
import FVS
Here we construct an example with an optimal solution. G2_FVS shoule be ['A'] as a list.
G2=nx.DiGraph()
G2.add_edges_from([('A','B'),('B','C'),('C','A'),('A','D'),('D','A')])
G2_FVS=FVS.FVS(G2)
Here we construct an example of three-node feedback loops.
We show how you change all the parameters and set a random seed.
Your result should be the same with the same randomseed.
G3=nx.DiGraph()
G3.add_edges_from([('A','B'),('B','C'),('C','A')])
G3_FVS=FVS.FVS(G3, T_0=0.6, alpha=0.99, maxMvt_factor=5, maxFail=50, randomseed=1)
The MIT License (MIT)
Copyright (c) 2017 Gang Yang and Reka Albert.
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
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