Abstract
The minimum feedback arc set problem asks to delete a minimum number of arcs (directed edges) from a digraph (directed graph) to make it free of any directed cycles. In this work we approach this fundamental cycle-constrained optimization problem by considering a generalized task of dividing the digraph into D layers of equal size. We solve the D-segmentation problem by the replica-symmetric mean field theory and belief-propagation heuristic algorithms. The minimum feedback arc density of a given random digraph ensemble is then obtained by extrapolating the theoretical results to the limit of large D. A divide-and-conquer algorithm (nested-BPR) is devised to solve the minimum feedback arc set problem with very good performance and high efficiency.
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Lan, Y., Mezić, I.: On the architecture of cell regulation networks. BMC Syst. Biol. 5, 37 (2011)
Gupte, M., Shankar, P., Li, J., Muthukrishnan, S., Iftode, L.: Finding hierarchy in directed online social networks. In: Proceedings of the twentieth International World Wide Web Conference, pp. 557–566 (Association for Computing Machinery, Hyderabad, India, 2011)
Xu, J., Lan, Y.: Hierarchical feedback modules and reaction hubs in cell signaling networks. PLoS ONE 10(5), e0125886 (2015)
Zhao, J.-H., Zhou, H.-J.: Feedback arcs and node hierarchy in directed networks. Chin. Phys. B 26, 078901 (2017)
Ispolatove, I., Maslov, S.: Detection of the dominant direction of information flow and feedback links in densely interconnected regulatory networks. BMC Bioinform. 9, 424 (2008)
Domínguez-García, V., Pigolotti, S., Muñoz, M.A.: Inherent directionality explains the lack of feedback loops in empirical networks. Sci. Rep. 4, 7497 (2014)
Karp, R.M.: Reducibility among combinatorial problems. In: Miller, E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Galinier, P., Lemamou, E., Bouzidi, M.W.: Applying local search to the feedback vertex set problem. J. Heuristics 19, 797–818 (2013)
Fu, Y., Anderson, P.W.: Application of statistical mechanics to np-complete problems in combinatorial optimisation. J. Phys. A 19, 1605–1620 (1986)
Mézard, M., Parisi, G.: Mean-field theory of randomly frustrated systems with finite connectivity. Europhys. Lett. 3, 1067–1074 (1987)
Sherrington, D., Wong, K.Y.M.: Graph bipartitioning and the bethe spin glass. J. Phys. A 20, L785–L791 (1987)
Lai, P.-Y., Goldschidt, Y.Y.: Application of statistical mechanics to combinatorial optimization problems: the chromatic number problem and \(q\)-partitioning of a graph. J. Stat. Phys. 48, 513–529 (1987)
Šulc, P., Zdeborová, L.: Belief propagation for graph partitioning. J. Phys. A 43, 285003 (2010)
Kawamoto, T., Kabashima, Y.: Limitations in the spectral method for graph partitioning: detectability threshold and localization of eigenvectors. Phys. Rev. E 91, 062803 (2015)
Mézard, M., Parisi, G.: The bethe lattice spin glass revisited. Eur. Phys. J. B 20, 217–233 (2001)
Yedidia, J.S., Freeman, W.T., Weiss, Y.: Understanding belief propagation and its generalizations. Technical Reports Mitsubishi Electric Research Laboratories (2001)
Yedidia, J.S., Freeman, W.T., Weiss, Y.: Constructing free-energy approximations and generalized belief-propagation algorithms. IEEE Trans. Inf. Theory 51, 2282–2312 (2005)
Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford University Press, New York (2009)
Xiao, J.-Q., Zhou, H.J.: Partition function loop series for a general graphical model: free-energy corrections and message-passing equations. J. Phys. A 44, 425001 (2011)
Zhou, H.J., Wang, C.: Region graph partition function expansion and approximate free energy landscapes: theory and some numerical results. J. Stat. Phys. 148, 513–547 (2012)
Mori, R.: Loop calculus for non-binary alphabets using concepts from information genometry. IEEE Tran. Inf. Theory 61, 1887–1904 (2015)
Zhou, H.-J.: Spin Glass and Message Passing. Science Press, Beijing (2015)
Zhou, H.-J.: Spin glass approach to the feedback vertex set problem. Eur. Phys. J. B 86, 455 (2013)
Bau, S., Wormald, N.C., Zhou, S.: Decycling numbers of random regular graphs. Random Struct. Algorithms 21, 397–413 (2002)
Haxell, P., Pikhurko, O., Thomason, A.: Maximum acyclic and fragmented sets in regular graphs. J. Graph Theory 57, 149–156 (2008)
Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)
Marino, R., Parisi, G., Ricci-Tersenghi, F.: The backtracking survey propagation algorithm for solving random \(k\)-sat problems. Nat. Commun. 7, 12996 (2016)
Braunstein, A., Zecchina, R.: Learning by message passing in networks of discrete synapses. Phys. Rev. Lett. 96, 030201 (2006)
Zhou, H.-J.: A spin glass approach to the directed feedback vertex set problem. J. Stat. Mech. 2016, 12 (2016)
Bauke, H., Mertens, S.: Random numbers for large-scale distributed monte carlo simulations. Phys. Rev. E 75, 066701 (2007)
Acknowledgements
We thank Dr. Jin-Hua Zhao for an earlier collaboration which stimulated the present Project, and thank Dr. Heiko Bauke for a helpful correspondence on the TRNG library of random number generators [30] which was called in our computer simulations. One of the authors (HJZ) acknowledges the hospitality of the Asia Pacific Center for Theoretical Physics (APCTP, Pohang, Korea) where the theoretical part of this Project was carried out during a short visit in November 2016. This research was partially supported by the National Natural Science Foundation of China (Grant Numbers 11121403 and 11647601).
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Xu, YZ., Zhou, HJ. Optimal Segmentation of Directed Graph and the Minimum Number of Feedback Arcs. J Stat Phys 169, 187–202 (2017). https://doi.org/10.1007/s10955-017-1860-5
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DOI: https://doi.org/10.1007/s10955-017-1860-5