Settling the Complexity of Local Max-Cut (Almost) Completely

  • Robert Elsässer
  • Tobias Tscheuschner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schäffer and Yannakakis (SICOMP, 1991) showed \(\mathcal{PLS}\)-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n 2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Röglin, and Vöcking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local Max-Cut problem with FLIP-neighborhood is \(\mathcal{PLS}\)-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is \(\mathcal{PLS}\)-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless \(\mathcal{PLS}\)\(\mathcal{P}\)).

On the other side, we also prove that on graphs with degree O(logn) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance σ 2, every FLIP local search terminates in time polynomial in n and σ − 1, with probability 1 − n − Ω(1). Putting both results together, we may conclude that although local Max-Cut is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.


Max-Cut PLS graphs local search smoothed complexity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arthur, D., Manthey, B., Röglin, H.: k-Means has polynomial smoothed complexity. In: FOCS 2009, pp. 405–414 (2009)Google Scholar
  2. 2.
    Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. Journal of the ACM (JACM) 55(6), art. 25 (2008)Google Scholar
  3. 3.
    Blum, A., Dunagan, J.: Smoothed analysis of the perceptron algorithm for linear programming. In: SODA, pp. 905–914 (2002)Google Scholar
  4. 4.
    Beier, R., Vöcking, B.: Typical properties of winners and losers in discrete optimization. In: STOC, pp. 343–352 (2004)Google Scholar
  5. 5.
    Englert, M., Röglin, H., Vöcking, B.: Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP. In: SODA, pp. 1295–1304 (2006)Google Scholar
  6. 6.
    Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure Nash Equilibria. In: STOC, pp. 604–612 (2004)Google Scholar
  7. 7.
    Gairing, M., Savani, R.: Computing stable outcomes in hedonic games. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds.) Algorithmic Game Theory. LNCS, vol. 6386, pp. 174–185. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and intractability, a guide to the theory of NP-completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  9. 9.
    Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: How easy is local search? Journal of Computer and System Sciences 37(1), 79–100 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Krentel, M.W.: Structure in locally optimal solutions. In: FOCS, pp. 216–221 (1989)Google Scholar
  11. 11.
    Kelner, J.A., Nikolova, E.: On the hardness and smoothed complexity of quasi-concave minimization. In: FOCS, pp. 472–482 (2007)Google Scholar
  12. 12.
    Loebl, M.: Efficient maximal cubic graph cuts. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 351–362. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  13. 13.
    Monien, B., Dumrauf, D., Tscheuschner, T.: Local search: Simple, successful, but sometimes sluggish. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 1–17. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Monien, B., Tscheuschner, T.: On the power of nodes of degree four in the local max-cut problem. In: Calamoneri, T., Diaz, J. (eds.) CIAC 2010. LNCS, vol. 6078, pp. 264–275. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Poljak, S.: Integer linear programs and local search for max-cut. SIAM Journal on Computing 21(3), 450–465 (1995)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Poljak, S., Tuza, Z.: Maximum cuts and largest bipartite subgraphs. Combinatorial Optimization, pp. 181–244. American Mathematical Society, Providence (1995)zbMATHGoogle Scholar
  17. 17.
    Röglin, H.: Personal communication (2010)Google Scholar
  18. 18.
    Röglin, H., Vöcking, B.: Smoothed analysis of integer programming. Math. Program. 110(1), 21–56 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Spielmann, D., Teng, S.-H.: Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM (JACM) 51(3), 385–463 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Spielmann, D., Teng, S.-H.: Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Commun. ACM 52(10), 76–84 (2009)CrossRefGoogle Scholar
  21. 21.
    Schäffer, A.A., Yannakakis, M.: Simple local search problems that are hard to solve. SIAM Journal on Computing 20(1), 56–87 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Vershynin, R.: Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method. In: FOCS, pp. 133–142 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Robert Elsässer
    • 1
  • Tobias Tscheuschner
    • 1
  1. 1.Faculty of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornGermany

Personalised recommendations