Skip to main content

Computing Minimum Cuts by Randomized Search Heuristics

Abstract

We study the minimum s-t-cut problem in graphs with costs on the edges in the context of evolutionary algorithms. Minimum cut problems belong to the class of basic network optimization problems that occur as crucial subproblems in many real-world optimization problems and have a variety of applications in several different areas. We prove that there exist instances of the minimum s-t-cut problem that cannot be solved by standard single-objective evolutionary algorithms in reasonable time. On the other hand, we develop a bi-criteria approach based on the famous maximum-flow minimum-cut theorem that enables evolutionary algorithms to find an optimal solution in expected polynomial time.

References

  1. Baier, G.: Flows with path restrictions. Ph.D. Thesis, TU Berlin (2003)

  2. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)

    MATH  Google Scholar 

  3. Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)

    MATH  Article  MathSciNet  Google Scholar 

  4. Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)

    MATH  Article  MathSciNet  Google Scholar 

  5. Duarte, A., Sánchez, Á., Fernández, F., Cabido, R.: A low-level hybridization between memetic algorithm and VNS for the max-cut problem. In: Proc. of GECCO ’05, pp. 999–1006 (2005)

  6. Friedrich, T., He, J., Hebbinghaus, N., Neumann, F., Witt, C.: Approximating covering problems by randomized search heuristics using multi-objective models. In: Proc. of GECCO ’07, pp. 797–804 (2007)

  7. Giel, O.: Expected runtimes of a simple multi-objective evolutionary algorithm. In: Proc. of CEC ’03, pp. 1918–1925. IEEE Press, New York (2003)

    Google Scholar 

  8. Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Proc. of STACS ’03, pp. 415–426 (2003)

  9. Horoba, C., Neumann, F.: Benefits and drawbacks for the use of epsilon-dominance in evolutionary multi-objective optimization. In: Proc. of GECCO ’08, pp. 641–648 (2008)

  10. Jansen, T., Wegener, I.: Evolutionary algorithms—how to cope with plateaus of constant fitness and when to reject strings of the same fitness. IEEE Trans. Evol. Comput. 5(6), 589–599 (2001)

    Article  Google Scholar 

  11. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 3rd edn. Springer, Berlin (2005)

    Google Scholar 

  12. Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diversity in evolutionary multiobjective optimization. Evol. Comput. 10(3), 263–282 (2002)

    Article  Google Scholar 

  13. Laumanns, M., Thiele, L., Zitzler, E.: Running time analysis of multiobjective evolutionary algorithms on pseudo-boolean functions. IEEE Trans. Evol. Comput. 8(2), 170–182 (2004)

    Article  Google Scholar 

  14. Liang, K.-H., Yao, X., Newton, C.S., Hoffman, D.: A new evolutionary approach to cutting stock problems with and without contiguity. Comput. Oper. Res. 29(12), 1641–1659 (2002)

    Article  MathSciNet  Google Scholar 

  15. Neumann, F.: Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem. Eur. J. Oper. Res. 181(3), 1620–1629 (2007)

    MATH  Article  Google Scholar 

  16. Neumann, F., Reichel, J.: Approximating minimum multicuts by evolutionary multi-objective algorithms. In: Proc. of PPSN’08, pp. 72–81 (2008)

  17. Neumann, F., Reichel, J., Skutella, M.: Computing minimum cuts by randomized search heuristics. In: Proc. of GECCO ’08, pp. 779–786 (2008)

  18. Neumann, F., Wegener, I.: Minimum spanning trees made easier via multi-objective optimization. Nat. Comput. 5(3), 305–319 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  19. Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theor. Comput. Sci. 378(1), 32–40 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  20. Puchinger, J., Raidl, G.R., Koller, G.: Solving a real-world glass cutting problem. In: Proc. of EvoCOP ’04, pp. 165–176. Springer, Berlin (2004)

    Google Scholar 

  21. Reichel, J., Skutella, M.: Evolutionary algorithms and matroid optimization problems. In: Proc. of GECCO ’07, pp. 947–954 (2007)

  22. Reichel, J., Skutella, M.: On the size of weights in randomized search heuristics. In: Proc. of FOGA ’09, pp. 21–28. ACM, New York (2009)

    Chapter  Google Scholar 

  23. Siarry, P., Michalewicz, Z. (eds.): Advances in Metaheuristics for Hard Optimization. Natural Computing Series. Springer, Berlin (2008)

    MATH  Google Scholar 

  24. Witt, C.: Worst-case and average-case approximations by simple randomized search heuristics. In: Proc. of STACS ’05, pp. 44–56 (2005)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Frank Neumann.

Additional information

A conference version appeared in GECCO 2008 [17].

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Neumann, F., Reichel, J. & Skutella, M. Computing Minimum Cuts by Randomized Search Heuristics. Algorithmica 59, 323–342 (2011). https://doi.org/10.1007/s00453-009-9370-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-009-9370-8

  • Evolutionary algorithms
  • Minimum s-t-cuts
  • Multi-objective optimization
  • Randomized search heuristics