Mathematical Programming

, Volume 118, Issue 1, pp 47–74 | Cite as

A simple GAP-canceling algorithm for the generalized maximum flow problem

FULL LENGTH PAPER

Abstract

We give a simple primal algorithm for the generalized maximum flow problem that repeatedly finds and cancels generalized augmenting paths (GAPs). We use ideas of Wallacher (A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms, 1991) to find GAPs that have a good trade-off between the gain of the GAP and the residual capacity of its arcs; our algorithm may be viewed as a special case of Wayne’s algorithm for the generalized minimum-cost circulation problem (Wayne in Math Oper Res 27:445–459, 2002). Most previous algorithms for the generalized maximum flow problem are dual-based; the few previous primal algorithms (including Wayne in Math Oper Res 27:445–459, 2002) require subroutines to test the feasibility of linear programs with two variables per inequality (TVPIs). We give an O(mn) time algorithm for finding negative-cost GAPs which can be used in place of the TVPI tester. This yields an algorithm with O(m log(mB/ε)) iterations of O(mn) time to compute an ε-optimal flow, or O(m 2 log (mB)) iterations to compute an optimal flow, for an overall running time of O(m 3 nlog(mB)). The fastest known running time for this problem is \(\tilde{O}(m^2n\log B)\) , and is due to Radzik (Theor Comput Sci 312:75–97, 2004), building on earlier work of Goldfarb et al. (Math Oper Res 22:793–802, 1997).

Mathematics Subject Classification (2000)

68Q25 90C35 90B10 05C85 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aspvall B. and Shiloach Y. (1990). A polynomial time algorithm for solving systems of linear inequalities with two variables per inequality. SIAM J. Comput. 9: 827–845 CrossRefMathSciNetGoogle Scholar
  2. 2.
    Aspvall, B.I.: Efficient Algorithms for Certain Satisfiability and Linear Programming Problems. PhD thesis, Department of Computer Science, Stanford University, August 1980. Also appears as Technical Report STAN-CS-80-822Google Scholar
  3. 3.
    Bertsekas D.P. and Tseng P. (1998). Relaxation methods for minimum cost ordinary and generalized network flow problems. Oper. Res. 36: 93–114 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Cherkassky B.V. and Goldberg A.V. (1999). Negative-cycle detection algorithms. Math. Programm. 85: 277–311 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen E. and Megiddo N. (1994). Improved algorithms for linear inequalities with two variables per inequality. SIAM J. Comput. 23: 1313–1347 MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cohen E. and Megiddo N. (1994). New algorithms for generalized network flows. Math. Programm. 64: 325–336 CrossRefMathSciNetGoogle Scholar
  7. 7.
    Cormen T.H., Leiserson C.E. and Rivest R.L. (1990). Introduction to Algorithms. MIT Press, Cambridge MATHGoogle Scholar
  8. 8.
    Dantzig G.B. (1963). Linear Programming and Extensions. Princeton University Press, Princeton MATHGoogle Scholar
  9. 9.
    Dash Optimization. XPRESS-MP 2004D, 2004Google Scholar
  10. 10.
    Edmonds J. and Karp R.M. (1972). Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19: 248–264 MATHCrossRefGoogle Scholar
  11. 11.
    Fleischer L.K. and Wayne K.D. (2002). Fast and simple approximation schemes for generalized flow. Math. Programm. 91: 215–238 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Garg, N., Könemann, J.: Faster and simpler algorithms for multicommodity flow and other fractional packing problems. In: Proceedings of the 39th IEEE Symposium on the Foundations of Computer Science, pp 300–309 (1998)Google Scholar
  13. 13.
    Goldberg A.V., Plotkin S.A. and Tardos É. (1991). Combinatorial algorithms for the generalized circulation problem. Math. Oper. Res. 16: 351–381 MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Goldberg A.V. and Tarjan R.E. (1989). Finding minimum-cost circulations by canceling negative cycles. J. ACM 36: 873–886 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Goldfarb D. and Jin Z. (1996). A faster combinatorial algorithm for the generalized circulation problem. Math. Oper. Res. 21: 529–539 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Goldfarb D., Jin Z. and Lin Y. (2002). A polynomial dual simplex algorithm for the generalized circulation problem. Math. Programm. 91: 271–288 MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Goldfarb D., Jin Z. and Orlin J.B. (1997). Polynomial-time highest-gain augmenting path algorithms for the generalized circulation problem. Math. Oper. Res. 22: 793–802 MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gondran M. and Minoux M. (1984). Graphs and Algorithms. Wiley, New York MATHGoogle Scholar
  19. 19.
    Grinold R.C. (1973). Calculating maximal flows in a network with positive gains. Oper. Res. 21: 528–541 MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hochbaum D.S. and Naor J. (1994). Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J. Comput. 23: 1179–1192 MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    ILOG. CPLEX 9.0, 2003Google Scholar
  22. 22.
    Jensen P.A. and Bhaumik G. (1977). A flow augmentation approach to the network with gain minimum cost flow problem. Management Science 23: 631–643 MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Jewell W.S. (1962). Optimal flow through networks with gains. Oper. Res. 10: 476–499 MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kleinberg J. and Tardos É. (2006). Algorithm Design. Addison-Wesley, Reading Google Scholar
  25. 25.
    Klingman D., Napier A. and Stutz J. (1974). NETGEN: a program for generating large scale capacitated assignment, transportation, and minimum cost flow network problems. Manage. Sci. 20: 814–821 MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Maurras J.F. (1972). Optimization of the flow through networks with gains. Math. Programm. 3: 135–144 MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Minieka E. (1972). Optimal flow in a network with gains. INFOR 10: 171–178 MATHGoogle Scholar
  28. 28.
    Oldham J.D. (2001). Combinatorial approximation algorithms for generalized flow problems. J. Algorithms 38: 135–169 MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Onaga K. (1967). Optimum flows in general communications networks. J. Franklin Inst. 283: 308–327 MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Radzik T. (1998). Faster algorithms for the generalized network flow problem. Math. Oper. Res. 23: 69–100 MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Radzik T. (2004). Improving time bounds on maximum generalised flow computations by contracting the network. Theor. Comput. Sci. 312: 75–97 MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Radzik T. and Yang S. (2001). Experimental evaluation of algorithmic solutions for the maximum generalised flow problem. Technical Report TR-01-09, Department of Computer Science, King’s College, London Google Scholar
  33. 33.
    Tardos É. and Wayne K.D. (1998). Simple generalized maximum flow algorithms. In: Bixby, R.E., Boyd, E.A. and Ríos-Mercado, R.Z. (eds) Integer Programming and Combinatorial Optimization, number 1412 in Lecture Notes in Computer Science, pp 310–324. Springer, New York CrossRefGoogle Scholar
  34. 34.
    Truemper, K.: Optimal Flows in Networks with Positive Gains. PhD thesis, Department of Operations Research, Case Western Reserve University, May 1973Google Scholar
  35. 35.
    Truemper K. (1977). On max flows with gains and pure min-cost flows. SIAM J. Appl. Math. 32: 450–456 MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Tseng P. and Bertsekas D.P. (2000). An ε-relaxation method for separable convex cost generalized network flow problems. Math. Programm. 88: 85–104 MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Wallacher, C.: A generalization of the minimum-mean cycle selection rule in cycle canceling algorithms. Technical report, Institut für Angewandte Mathematik, Technische Universität Carolo-Wilhelmina, November 1991Google Scholar
  38. 38.
    Wayne K.D. (2002). A polynomial combinatorial algorithm for generalized minimum cost flow. Math. Oper. Res. 27: 445–459 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Center for Applied MathematicsCornell UniversityIthacaUSA
  2. 2.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

Personalised recommendations