The Partial Augment–Relabel Algorithm for the Maximum Flow Problem

  • Andrew V. Goldberg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5193)

Abstract

The maximum flow problem is a classical optimization problem with many applications. For a long time, HI-PR, an efficient implementation of the highest-label push-relabel algorithm, has been a benchmark due to its robust performance. We propose another variant of the push-relabel method, the partial augment-relabel (PAR) algorithm. Our experiments show that PAR is very robust. It outperforms HI-PR on all problem families tested, asymptotically in some cases.

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References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Anderson, R.J., Setubal, J.C.: Goldberg’s Algorithm for the Maximum Flow in Perspective: a Computational Study. In: Johnson, D.S., McGeoch, C.C. (eds.) Network Flows and Matching: First DIMACS Implementation Challenge, pp. 1–18. AMS (1993)Google Scholar
  3. 3.
    Babenko, M.A., Goldberg, A.V.: Experimental Evaluation of a Parametric Flow Algorithm. Technical Report MSR-TR-2006-77, Microsoft Research (2006)Google Scholar
  4. 4.
    Badics, T., Boros, E.: Implementing a Maximum Flow Algorithm: Experiments with Dynamic Trees. In: Johnson, D.S., McGeoch, C.C. (eds.) Network Flows and Matching: First DIMACS Implementation Challenge, pp. 65–96. AMS (1993)Google Scholar
  5. 5.
    Boykov, Y., Kolmogorov, V.: An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision. IEEE transactions on Pattern Analysis and Machine Intelligence 26(9), 1124–1137 (2004)CrossRefGoogle Scholar
  6. 6.
    Boykov, Y., Veksler, O.: Graph Cuts in Vision and Graphics: Theories and Applications. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision, pp. 109–131. Springer, Heidelberg (2006)Google Scholar
  7. 7.
    Chandran, B., Hochbaum, D.: A computational study of the pseudoflow and push-relabel algorithms for the maximum flow problem (submitted, 2007)Google Scholar
  8. 8.
    Chekuri, C.S., Goldberg, A.V., Karger, D.R., Levine, M.S., Stein, C.: Experimental Study of Minimum Cut Algorithms. In: Proc. 8th ACM-SIAM Symposium on Discrete Algorithms, pp. 324–333 (1997)Google Scholar
  9. 9.
    Cheriyan, J., Maheshwari, S.N.: Analysis of Preflow Push Algorithms for Maximum Network Flow. SIAM J. Comput. 18, 1057–1086 (1989)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cherkassky, B.V.: A Fast Algorithm for Computing Maximum Flow in a Network. In: Karzanov, A.V. (ed.) Collected Papers. Combinatorial Methods for Flow Problems, vol. 3, pp. 90–96. The Institute for Systems Studies, Moscow (1979) (in russian); English translation appears in AMS Trans., vol.158, pp. 23–30 (1994)Google Scholar
  11. 11.
    Cherkassky, B.V., Goldberg, A.V.: On Implementing Push-Relabel Method for the Maximum Flow Problem. Algorithmica 19, 390–410 (1997)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dantzig, G.B.: Application of the Simplex Method to a Transportation Problem. In: Koopmans, T.C. (ed.) Activity Analysis and Production and Allocation, pp. 359–373. Wiley, New York (1951)Google Scholar
  13. 13.
    Derigs, U., Meier, W.: An Evaluation of Algorithmic Refinements and Proper Data-Structures for the Preflow-Push Approach for Maximum Flow. In: ASI Series on Computer and System Sciences vol. 8, pp. 209–223. NATO (1992)Google Scholar
  14. 14.
    Dinic, E.A.: Metod porazryadnogo sokrashcheniya nevyazok i transportnye zadachi. In: Issledovaniya po Diskretnoĭ Matematike, Nauka, Moskva (1973) (in russian); Title translation: Excess Scaling and Transportation ProblemsGoogle Scholar
  15. 15.
    Edmonds, J., Karp, R.M.: Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems. J. Assoc. Comput. Mach. 19, 248–264 (1972)MATHGoogle Scholar
  16. 16.
    Ford, L.R., Fulkerson, D.R.: Maximal Flow Through a Network. Canadian Journal of Math. 8, 399–404 (1956)MATHMathSciNetGoogle Scholar
  17. 17.
    Goldberg, A.V.: Efficient Graph Algorithms for Sequential and Parallel Computers. PhD thesis, M.I.T., January 1987. Technical Report TR-374, Lab. for Computer Science, M.I.T (1987)Google Scholar
  18. 18.
    Goldberg, A.V.: An Efficient Implementation of a Scaling Minimum-Cost Flow Algorithm. J. Algorithms 22, 1–29 (1997)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Goldberg, A.V., Rao, S.: Beyond the Flow Decomposition Barrier. J. Assoc. Comput. Mach. 45, 753–782 (1998)MathSciNetGoogle Scholar
  20. 20.
    Goldberg, A.V., Tarjan, R.E.: A New Approach to the Maximum Flow Problem. J. Assoc. Comput. Mach. 35, 921–940 (1988)MATHMathSciNetGoogle Scholar
  21. 21.
    Goldfarb, D., Grigoriadis, M.D.: A Computational Comparison of the Dinic and Network Simplex Methods for Maximum Flow. Annals of Oper. Res. 13, 83–123 (1988)MathSciNetGoogle Scholar
  22. 22.
    Hagerup, T., Sanders, P., Träff, J.L̃.: An implementation of the binary blocking flow algorithm. Algorithm Engineering, 143–154 (1998)Google Scholar
  23. 23.
    Johnson, D.S., McGeoch, C.C.: Network Flows and Matching: First DIMACS Implementation Challenge. In: AMS. Proceedings of the 1-st DIMACS Implementation Challenge (1993)Google Scholar
  24. 24.
    Karzanov, A.V.: Determining the Maximal Flow in a Network by the Method of Preflows. Soviet Math. Dok. 15, 434–437 (1974)MATHGoogle Scholar
  25. 25.
    King, V., Rao, S., Tarjan, R.: A Faster Deterministic Maximum Flow Algorithm. J. Algorithms 17, 447–474 (1994)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Mehlhorn, K., Naher, S.: LEDA: A Platform for Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  27. 27.
    Nguyen, Q.C., Venkateswaran, V.: Implementations of Goldberg-Tarjan Maximum Flow Algorithm. In: Johnson, D.S., McGeoch, C.C. (eds.) Network Flows and Matching: First DIMACS Implementation Challenge, pp. 19–42. AMS (1993)Google Scholar
  28. 28.
    Ogielski, A.T.: Integer Optimization and Zero-Temperature Fixed Point in Ising Random-Field Systems. Phys. Rev. Lett. 57, 1251–1254 (1986)CrossRefGoogle Scholar
  29. 29.
    Sleator, D.D., Tarjan, R.E.: A Data Structure for Dynamic Trees. J. Comput. System Sci. 26, 362–391 (1983)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Tuncel, L.: On the Complexity of Preflow-Push Algorithms for Maximum-Flow Problems. Algorithmica 11, 353–359 (1994)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andrew V. Goldberg
    • 1
  1. 1.Microsoft Research – Silicon ValleyMountain View

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