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

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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).

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References

  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

    Article  MathSciNet  Google Scholar 

  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-80-822

  3. Bertsekas D.P. and Tseng P. (1998). Relaxation methods for minimum cost ordinary and generalized network flow problems. Oper. Res. 36: 93–114

    Article  MathSciNet  Google Scholar 

  4. Cherkassky B.V. and Goldberg A.V. (1999). Negative-cycle detection algorithms. Math. Programm. 85: 277–311

    Article  MATH  MathSciNet  Google Scholar 

  5. Cohen E. and Megiddo N. (1994). Improved algorithms for linear inequalities with two variables per inequality. SIAM J. Comput. 23: 1313–1347

    Article  MATH  MathSciNet  Google Scholar 

  6. Cohen E. and Megiddo N. (1994). New algorithms for generalized network flows. Math. Programm. 64: 325–336

    Article  MathSciNet  Google Scholar 

  7. Cormen T.H., Leiserson C.E. and Rivest R.L. (1990). Introduction to Algorithms. MIT Press, Cambridge

    MATH  Google Scholar 

  8. Dantzig G.B. (1963). Linear Programming and Extensions. Princeton University Press, Princeton

    MATH  Google Scholar 

  9. Dash Optimization. XPRESS-MP 2004D, 2004

  10. Edmonds J. and Karp R.M. (1972). Theoretical improvements in algorithmic efficiency for network flow problems. J. ACM 19: 248–264

    Article  MATH  Google Scholar 

  11. Fleischer L.K. and Wayne K.D. (2002). Fast and simple approximation schemes for generalized flow. Math. Programm. 91: 215–238

    Article  MATH  MathSciNet  Google Scholar 

  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)

  13. Goldberg A.V., Plotkin S.A. and Tardos É. (1991). Combinatorial algorithms for the generalized circulation problem. Math. Oper. Res. 16: 351–381

    Article  MATH  MathSciNet  Google Scholar 

  14. Goldberg A.V. and Tarjan R.E. (1989). Finding minimum-cost circulations by canceling negative cycles. J. ACM 36: 873–886

    Article  MATH  MathSciNet  Google Scholar 

  15. Goldfarb D. and Jin Z. (1996). A faster combinatorial algorithm for the generalized circulation problem. Math. Oper. Res. 21: 529–539

    Article  MATH  MathSciNet  Google Scholar 

  16. Goldfarb D., Jin Z. and Lin Y. (2002). A polynomial dual simplex algorithm for the generalized circulation problem. Math. Programm. 91: 271–288

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  18. Gondran M. and Minoux M. (1984). Graphs and Algorithms. Wiley, New York

    MATH  Google Scholar 

  19. Grinold R.C. (1973). Calculating maximal flows in a network with positive gains. Oper. Res. 21: 528–541

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  21. ILOG. CPLEX 9.0, 2003

  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

    Article  MATH  MathSciNet  Google Scholar 

  23. Jewell W.S. (1962). Optimal flow through networks with gains. Oper. Res. 10: 476–499

    Article  MATH  MathSciNet  Google Scholar 

  24. Kleinberg J. and Tardos É. (2006). Algorithm Design. Addison-Wesley, Reading

    Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  26. Maurras J.F. (1972). Optimization of the flow through networks with gains. Math. Programm. 3: 135–144

    Article  MATH  MathSciNet  Google Scholar 

  27. Minieka E. (1972). Optimal flow in a network with gains. INFOR 10: 171–178

    MATH  Google Scholar 

  28. Oldham J.D. (2001). Combinatorial approximation algorithms for generalized flow problems. J. Algorithms 38: 135–169

    Article  MATH  MathSciNet  Google Scholar 

  29. Onaga K. (1967). Optimum flows in general communications networks. J. Franklin Inst. 283: 308–327

    Article  MATH  MathSciNet  Google Scholar 

  30. Radzik T. (1998). Faster algorithms for the generalized network flow problem. Math. Oper. Res. 23: 69–100

    Article  MATH  MathSciNet  Google Scholar 

  31. Radzik T. (2004). Improving time bounds on maximum generalised flow computations by contracting the network. Theor. Comput. Sci. 312: 75–97

    Article  MATH  MathSciNet  Google Scholar 

  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. 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

    Chapter  Google Scholar 

  34. Truemper, K.: Optimal Flows in Networks with Positive Gains. PhD thesis, Department of Operations Research, Case Western Reserve University, May 1973

  35. Truemper K. (1977). On max flows with gains and pure min-cost flows. SIAM J. Appl. Math. 32: 450–456

    Article  MATH  MathSciNet  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  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 1991

  38. Wayne K.D. (2002). A polynomial combinatorial algorithm for generalized minimum cost flow. Math. Oper. Res. 27: 445–459

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Mateo Restrepo

  2. School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 14853, USA

    David P. Williamson

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  1. Mateo Restrepo
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  2. David P. Williamson
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Correspondence to David P. Williamson.

Additional information

David P. Williamson is supported in part by an IBM Faculty Partnership Award and NSF grant CCF-0514628.

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Restrepo, M., Williamson, D.P. A simple GAP-canceling algorithm for the generalized maximum flow problem. Math. Program. 118, 47–74 (2009). https://doi.org/10.1007/s10107-007-0183-8

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