Annals of Combinatorics

, Volume 19, Issue 2, pp 373–396 | Cite as

Feedback Arc Set Problem and NP-Hardness of Minimum Recurrent Configuration Problem of Chip-Firing Game on Directed Graphs

  • Kévin Perrot
  • Trung Van Pham


In this paper we present further studies of recurrent configurations of chip-firing games on Eulerian directed graphs (simple digraphs), a class on the way from undirected graphs to general directed graphs. A computational problem that arises naturally from this model is to find the minimum number of chips of a recurrent configuration, which we call the minimum recurrent configuration (MINREC) problem. We point out a close relationship between MINREC and the minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and prove that both problems are NP-hard.


chip-firing game critical configuration recurrent configuration Eulerian digraph feedback arc set complexity sandpile model 

Mathematics Subject Classification

68R10 90C27 05C20 05C45 


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  1. 1.
    Bak P., Tang C., Wiesenfeld K.: Self-organized criticality: an explanation of the 1/ f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Benson B., Chakrabarty D., Tetali P.: G-parking functions, acyclic orientations and spanning trees. Discrete Math. 310(8), 1340–1353 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Biggs N.L.: Chip-firing and the critical group of a graph. J. Algebraic Combin. 9(1), 25–45 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Björner A., Lovász L.: Chip-firing games on directed graphs. J. Algebraic Combin. 1(4), 305–328 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Björner A., Lovász L., Shor P.W.: Chip-firing games on graphs. European J. Combin. 12(4), 283–291 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Borobia A., Nutov Z., Penn M.: Doubly stochastic matrices and dicycle covers and packings in Eulerian digraphs. Linear Algebra Appl. 246, 361–371 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Charbit P., Thomassé S., Yeo A.: The minimum feedback arc set problem is NP-hard for tournaments. Combin. Probab. Comput. 16(1), 1–4 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chebikin D., Pylyavskyy P.: A family of bijections between G-parking functions and spanning trees. J. Combin. Theory Ser. A 110(1), 31–41 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cori R., Le Borgne Y.: The sand-pile model and Tutte polynomials. Adv. Appl. Math. 30(1-2), 44–52 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dhar D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Flier, H.-F.R.: Optimization of railway operations. PhD thesis, ETH, Zürich. Avaible at: (2011)
  12. 12.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Grad. Texts in Math., Vol. 207. Springer-Verlag, New York (2001)Google Scholar
  13. 13.
    Goemans, M.X., Williamson, D.P.: Primal-dual approximation algorithms for feedback problems in planar graphs. In: Cunningham, W.H., McCormick, S.T., Queyranne, M. (eds.) Integer Programming and Combinatorial Optimization, pp. 147–161. Springer, Berlin (1996)Google Scholar
  14. 14.
    Greene C., Zaslavsky T.: On the interpretation ofWhitney numbers through arrangement of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. Trans. Amer. Math. Soc. 280(1), 97–126 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Guo J., Hüffner F., Moser H.: Feedback arc set in bipartite tournaments is NP-complete. Inform. Process. Lett. 102(2-3), 62–65 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Guzmán, J., Klivans, C.: Chip-firing and energy minimization on M-matrices. arXiv:1403.1635 (2014)
  17. 17.
    Holroyd, A.E., Levine, L., Mészáros, K., Peres, Y., Propp, J., Wilson, D.B.: Chip-firing and rotor-routing on directed graphs. In: Sidoravicius, V., Vares, M.E. (eds.) In and Out of Equilibrium II, pp. 331–364. Birkhäuser, Basel (2008)Google Scholar
  18. 18.
    Huang, H., Ma, J., Shapira, A., Sudakov, B., Yuster, R.: Large feedback arc sets, high minimum degree subgraphs, and long cycles in Eulerian digraphs. SubmittedGoogle Scholar
  19. 19.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)Google Scholar
  20. 20.
    Latapy M., Phan H.D.: The lattice structure of chip firing games and related models. Phys. D 155(1-2), 69–82 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Magnien C.: Classes of lattices induced by chip firing (and sandpile) dynamics. European J. Combin. 24(6), 665–683 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Majumdar S.N., Dhar D.: Equivalence between the Abelian sandpile model and the q → 0 limit of the Potts model. Phys. A 185, 129–145 (1992)CrossRefGoogle Scholar
  23. 23.
    Merino López C.: Chip firing and the Tutte polynomial. Ann. Combin. 1(3), 253–259 (1997)CrossRefzbMATHGoogle Scholar
  24. 24.
    Perkinson, D., Perlman, J., Wilmes, J.: Primer for the algebraic geometry of sandpiles. In: Amini, O., Baker, M., Faber, X. (eds.) Tropical and Non-Archimedean Geometry, pp. 211–256. Amer. Math. Soc., Providence, RI (2013)Google Scholar
  25. 25.
    Perrot, K., Pham, T.V.: Chip-firing game and partial Tutte polynomial for Eulerian digraphs. arXiv:1306.0294 (2013)
  26. 26.
    Pham T.V., Phan T.H.D.: Lattices generated by chip firing game models: criteria and recognition algorithms. European J. Combin. 34(5), 812–832 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Postnikov A., Shapiro B.: Trees, parking functions, syzygies, and deformations of monomial ideals. Trans. Amer. Math. Soc. 356(8), 3109–3142 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Ramachandran V.: Finding a minimum feedback arc set in reducible flow graphs. J. Algorithms 9(3), 299–313 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Schulz M.: An NP-complete problem for the Abelian sandpile model. Complex Systems 17(1-2), 17–28 (2007)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Schulz, M.: Minimal recurrent configurations of chip firing games and directed acyclic graphs. Discrete Math. Theor. Comput. Sci. Proc. AL, 111–124 (2010)Google Scholar
  31. 31.
    Seymour P.D.: Packing directed circuits fractionally. Combinatorica 15(2), 281–288 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Seymour P.D.: Packing circuits in Eulerian digraphs. Combinatorica 16(2), 223–231 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Speer E.R.: Asymmetric Abelian sandpile models. J. Statist. Phys. 71(1-2), 61–74 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Stamm, H.: On feedback problems in planar digraphs. In: Möhring, R.H. (ed.) Graph-Theoretic Concepts in Computer Science. Lect. Notes Comput. Sci. Eng., Vol. 484, pp. 79–89. Springer, Berlin (1991)Google Scholar
  35. 35.
    Stanley, R.P.: Enumerative Combinatorics. Vol. 2. Cambridge University Press, Cambridge (1999)Google Scholar

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Departamento de Ingeniería MatemáticaUniversidad de Chile, CMM (UMI 2807 - CNRS)Blanco EncaldaChile
  2. 2.Aix Marseille UniversitéCNRSMarseilleFrance
  3. 3.Department of Mathematics for Computer ScienceInstitute of Mathematics, VASTHanoiVietnam

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