Mathematical Programming

, Volume 135, Issue 1–2, pp 337–367 | Cite as

Structural and algorithmic properties for parametric minimum cuts

  • Frieda Granot
  • S. Thomas McCormick
  • Maurice Queyranne
  • Fabio Tardella
Full Length Paper Series A

Abstract

We consider the minimum s, t-cut problem in a network with parametrized arc capacities. Following the seminal work of Gallo et al. (SIAM J. Comput. 18(1):30–55, 1989), classes of this parametric problem have been shown to enjoy the nice Structural Property that minimum cuts are nested, and the nice Algorithmic Property that all minimum cuts can be computed in the same asymptotic time as a single minimum cut by using a clever Flow Update step to move from one value of the parameter to the next. We present a general framework for parametric minimum cuts that extends and unifies such results. We define two conditions on parametrized arc capacities that are necessary and sufficient for (strictly) decreasing differences of the parametric cut function. Known results in parametric submodular optimization then imply the Structural Property. We show how to construct appropriate Flow Updates in linear time under the above conditions, implying that the Algorithmic Property also holds under these conditions. We then consider other classes of parametric minimum cut problems, without decreasing differences, for which we establish the Structural and/or the Algorithmic Property, as well as other cases where nested minimum cuts arise.

Keywords

Submodularity Minimum cut/maximum flow Parametric optimization Nested solutions Comparative statics 

Mathematics Subject Classification (2000)

90C35 90C31 

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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)MATHGoogle Scholar
  2. 2.
    Ahuja R.K., Orlin J.B., Stein C., Tarjan R.E.: Improved algorithms for bipartite network flow. SIAM J. Comput. 23, 906–933 (1994)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arai T., Ueno S., Kajitani Y.: Generalization of a theorem on the parametric maximum flow problem. Discret. Appl. Math. 41, 69–74 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Babenko, M., Derryberry, J., Goldberg, A.V., Tarjan, R.E., Zhou, Y.: Experimental evaluation of parametric maximum flow algorithms. In: Demetrescu, C. (ed.) Experimental and Efficient Algorithms 6th International Workshop, WEA 2007, Lecture Notes in Computer Science, vol. 4525, pp. 256–269. Springer (2007)Google Scholar
  5. 5.
    Ball M.O., Colbourn C.J., Provan J.S.: Network reliability. In: Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds) Handbook of Operations Research and Management Science vol 7: Network Models., pp. 673–762. North-Holland, Amsterdam (1995)Google Scholar
  6. 6.
    Brumelle, S., Granot, D., Liu, L.: An Extended Economic Selection Problem. Tech. report, Sauder School of Business, UBC, Vancouver (1995)Google Scholar
  7. 7.
    Brumelle S., Granot D., Liu L.: Ordered optimal solutions and parametric minimum cut problems. Discret. Optim. 2, 123–134 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Carstensen P.J.: Complexity of some parametric integer and network programming problems. Math. Program. 26, 64–75 (1983)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dinic E.A.: Algorithm for solution of a problem of maximum flow in networks with power estimation. Sov. Math. Dokl. 11, 1277–1280 (1970)Google Scholar
  10. 10.
    Dinkelbach W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Eisner M.J., Severance D.G.: Mathematical techniques for efficient record segmentation in large shared databases. J. ACM 23(4), 619–635 (1976)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fleischer L.K.: Universally maximum flow with piece-wise constant capacity functions. Networks 38, 1–11 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fleischer L.K., Iwata S.: A Push–Relabel framework for submodular function minimization and applications to parametric optimization. Discret. Appl. Math. 131, 311–322 (2003)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ford L.R. Jr, Fulkerson D.R.: Flows in Networks. Princeton University Press, Princeton (1962)MATHGoogle Scholar
  15. 15.
    Fujishige S.: Lexicographically optimal base of a polymatroid with respect to a weight vector. Math. Oper. Res. 5, 186–196 (1980)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Gallo G., Grigoriadis M.D., Tarjan R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18(1), 30–55 (1989)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Goldberg A.V., Rao S.: Beyond the flow decomposition barrier. J. ACM 45, 753–797 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Goldberg A.V., Tarjan R.E.: A new approach to the maximum flow problem. J. ACM 35, 921–940 (1988)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gusfield D., Martel C.: A fast algorithm for the generalized parametric minimum cut problem and applications. Algorithmica 7, 499–519 (1992)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Gusfield D., Tardos E.: A faster parametric minimum-cut algorithm. Algorithmica 11(3), 278–290 (1994)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hayrapetyan A., Kempe D., Pál M., Svitkina Z.: Unbalanced graph cuts. In: Brodal, G.S., Leonardi, S. (eds) ESA, Lecture Notes in Computer Science, vol. 3669, pp. 191–202. Springer, Berlin (2005)Google Scholar
  22. 22.
    Hochbaum D.S.: The pseudoflow algorithm: a new algorithm for the maximum flow problem. Oper. Res. 58, 992–1009 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Isbell J.R., Marlow W.H.: Attrition games. Naval Res. Logist. Q. 3, 71–94 (1956)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Karzanov A.V.: Determining the maximal flow in a network by the method of preflows. Sov. Math. Dokl. 15, 434–437 (1974)MATHGoogle Scholar
  25. 25.
    King V., Rao S., Tarjan R.E.: A faster deterministic maximum flow algorithm. J. Algorithms 17, 447–474 (1994)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Liu, L.: Ordered Optimal Solutions and Applications. Ph.D. thesis, The University of British Columbia (1996)Google Scholar
  27. 27.
    Martel C.: A comparison of phase and nonphase network flow algorithms. Networks 19(6), 691–705 (1989)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    McCormick S.T.: Fast algorithms for parametric scheduling come from extensions to parametric maximum flow. Oper. Res. 47(5), 744–756 (1999)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    McCormick S.T.: Submodular function minimization. In: Aardal, K., Nemhauser, G.L., Weismantel, R. (eds) Handbook of Discrete Optimization, North-Holland, Amsterdam (2004)Google Scholar
  30. 30.
    McCormick S.T., Ervolina T.R.: Computing maximum mean cuts. Discret. Appl. Math. 52, 53–70 (1994)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Milgrom P., Shannon C.: Monotone comparative statics. Econometrica 62(1), 157–180 (1994)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Nagano, K.: A Faster Parametric Submodular Function Minimization Algorithm and Applications. Tech. report, Department of Mathematical Informatics, University of Tokyo, Tokyo (2007)Google Scholar
  33. 33.
    Nagano, K.: On convex minimization over base polytopes. In: Fischetti, M., Williamson, D. (eds.), Proceedings of IPCO 12 (Ithaca, NY), pp. 252–266 (2007)Google Scholar
  34. 34.
    Ogier R.G.: Minimum-delay routing in continuous-time dynamic networks with piecewise-constant capacities. Networks 18, 303–318 (1988)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Ore Ø.: Theory of graphs, American Mathematical Society Colloquium Publications, vol. 38. American Mathematical Society, Providence (1962)Google Scholar
  36. 36.
    Orlin, J.B.: (2007) A faster strongly polynomial algorithm for submodular function minimization. In: Fischetti, M., Williamson, D. (eds.) Proceedings of IPCO 12 (Ithaca, NY), pp. 240–251 (2007)Google Scholar
  37. 37.
    Picard J.C., Queyranne M.: Selected applications of minimum cuts in networks. INFOR 20, 394–422 (1983)Google Scholar
  38. 38.
    Radzik T.: Parametric flows, weighted means of cuts, and fractional combinatorial optimization. In: Pardalos, P. (eds) Complexity in Numerical Optimization, pp. 351–386. World Scientific, Singapore (1993)CrossRefGoogle Scholar
  39. 39.
    Rhys J.M.W.: A selection problem of shared fixed costs and network flows. Manag. Sci 17, 200–207 (1970) (English)MATHCrossRefGoogle Scholar
  40. 40.
    Roosen J., Hennessy D.A.: Testing for the monotone likelihood ratio assumption. J. Bus. Econ. Stat. 22, 358–366 (2004)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Schrijver A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)MATHGoogle Scholar
  42. 42.
    Scutellà M.G.: A note on the parametric maximum flow problem and some related reoptimization issues. Ann. Oper. Res. 150, 231–244 (2007)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Shapley L.S.: On network flow functions. Naval Res. Logist. Q. 8, 151–158 (1961)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Stone H.S.: Critical load factors in two-processor distributed systems. IEEE Trans. Softw. Eng. 4(2), 254–258 (1978)MATHCrossRefGoogle Scholar
  45. 45.
    Tarjan R.E.: A simple version of Karzanov’s blocking flow algorithm. Oper. Res. Lett. 2(6), 265–268 (1984)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Topkis D.M.: Minimizing a submodular function on a lattice. Oper. Res. 26, 305–321 (1978)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Topkis D.M.: Supermodularity and Complementarity. Princeton University Press, Princeton (1998)Google Scholar
  48. 48.
    Topkis, D.H., Veinott, A.F. Jr.: Monotone solutions of extremal problems on lattices. In: 8th International Symposium on Mathematical Programming (Stanford University), p. 131 (1973)Google Scholar
  49. 49.
    Ward J., Zhang B., Jain S., Fry C., Olavson T., Mishal H., Amaral J., Beyer D., Brecht A., Cargille B., Chadinha R., Chou K., DeNyse G., Feng Q., Padovani C., Raj S., Sunderbruch K., Tarjan R., Venkatraman K., Woods J., Zhou J.: HP transforms product portfolio management with operations research. Interfaces 40, 17–32 (2010)CrossRefGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  • Frieda Granot
    • 1
  • S. Thomas McCormick
    • 1
  • Maurice Queyranne
    • 1
  • Fabio Tardella
    • 2
  1. 1.Sauder School of Business at UBCVancouverCanada
  2. 2.Sapienza University of RomeRomeItaly

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