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Theory of Computing Systems

, Volume 61, Issue 4, pp 987–1010 | Cite as

Minimum-Cost Flows in Unit-Capacity Networks

  • Andrew V. Goldberg
  • Sagi Hed
  • Haim Kaplan
  • Robert E. Tarjan
Article
  • 282 Downloads

Abstract

We consider combinatorial algorithms for the minimum-cost flow problem on networks with unit capacities, and special cases of the problem. Historically, researchers have developed special-purpose algorithms that exploit unit capacities. In contrast, for the maximum flow problem, the classical blocking flow and push-relabel algorithms for the general case also have the best bounds known for the special case of unit capacities. In this paper we show that the classical blocking flow push-relabel cost-scaling algorithms of Goldberg and Tarjan (Math. Oper. Res. 15, 430–466, 1990) for general minimum-cost flow problems achieve the best known bounds for unit-capacity problems as well. We also develop a cycle-canceling algorithm that extends Goldberg’s shortest path algorithm (Goldberg SIAM J. Comput. 24, 494–504, 1995) to minimum-cost, unit-capacity flow problems. Finally, we combine our ideas to obtain an algorithm that solves the minimum-cost bipartite matching problem in \(O(r^{1/2} m \log C)\) time, where m is the number of edges, C is the largest arc cost (assumed to be greater than 1), and r is the number of vertices on the small side of the vertex bipartition. This result generalizes (and simplifies) a result of Duan et al. (2011) and solves an open problem of Ramshaw and Tarjan (2012).

Keywords

Algorithms Minimum-cost flows Bipartite matching Assignment problem 

Notes

Acknowledgments

We thank an anonymous reviewer for a significant simplification of Step 4 of the refinement algorithm in Section 4.1.

Part of the work was done while Andrew V. Goldberg was at Microsoft Research.

Sagi Hed research supported by the Israel Science Foundation grants no. 822-10 and 1841/14, and the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11).

Haim Kaplan research supported by the Israel Science Foundation grants no. 822-10 and 1841/14, the German-Israeli Foundation for Scientific Research Development (GIF) grant no. 1161/2011, the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11).

References

  1. 1.
    Ahuja, R.K., Goldberg, A.V., Orlin, J.B., Tarjan, R.E.: Finding Minimum-Cost Flows by Double Scaling. Math. Prog. 53, 243–266 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. In: Nemhauser, G.L., Rinnooy Kan, A.H.G., Todd, M.J. (eds.) Optimization. Handbooks in operations research and management science, vol. 1, pp. 211–369. North-Holland, Amsterdam (1989)Google Scholar
  3. 3.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall (1993)Google Scholar
  4. 4.
    Bertsekas, D.P.: Distributed asynchronous relaxation methods for linear network flow problems. Technical Report LIDS-p-1986, Lab. for decision systems M.I.T. (1986)Google Scholar
  5. 5.
    Bland, R.G., Jensen, D.L.: On the computational behavior of a polynomial-time network flow algorithm. Math. Prog. 54, 1–41 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cherkassky, B.V., Goldberg, A.V., Silverstein, C.: Buckets, heaps, lists, and monotone priority queues. SIAM J. Comput. 28, 1326–1346 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cohen, M.B., Madry, A., Sankowski, P., Vladu, A.: Negative-weight shortest paths and unit capacity minimum cost flow in Õ (m 10/7 w) time. In: SODA, pp. 752–771 (2017)Google Scholar
  8. 8.
    Daitch, S.I., Spielman, D.A.: Faster approximate lossy generalized flow via interior point algorithms. In: STOC, pp. 451–460 (2008)Google Scholar
  9. 9.
    Duan, R., Pettie, S., Su, H.: Scaling algorithms for approximate and exact maximum weight matching. CoRR, arXiv:1112.0790 (2011)
  10. 10.
    Even, S., Tarjan, R.E.: Network flow and testing graph connectivity. SIAM J. Comput. 4, 507–518 (1975)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. Assoc. Comput. Mach. 34, 596–615 (1987)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gabow, H.N.: Scaling algorithms for network problems. J. Comp. Sys. Sci. 31, 148–168 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gabow, H.N., Tarjan, R.E.: Faster Scaling Algorithms for Network Problems. SIAM J. Comput. 18, 1013–1036 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Goldberg, A.V.: Scaling algorithms for the shortest paths problem. SIAM J. Comput. 24, 494–504 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Goldberg, A.V., Kennedy, R.: Global Price Updates Help. SIAM J. Disc. Math. 10, 551–572 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Goldberg, A.V., Tarjan, R.E.: Finding minimum-cost circulations by successive approximation. Math. Oper. Res. 15, 430–466 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hopcroft, J.E., Karp, R.M.: An n 5/2, Algorithm for Maximum Matching in Bipartite Graphs. SIAM J. Comput. 2, 225–231 (1973)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Karzanov, A.V.: O nakhozhdenii maksimal?nogo potoka v setyakh spetsial?nogo vida i nekotorykh prilozheniyakh. In: Matematicheskie Voprosy Upravleniya Proizvodstvom, volume 5. Moscow State University Press, Moscow, 1973. In Russian; title translation: On Finding Maximum Flows in Networks with Special Structure and Some ApplicationsGoogle Scholar
  19. 19.
    Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Reinhart, and Winston, New York (1976)MATHGoogle Scholar
  20. 20.
    Madry, A.: Navigating central path with electrical flows: From flows to matchings, and back. In: FOCS, pp. 253–262 (2013)Google Scholar
  21. 21.
    Orlin, J.B.: Faster strongly polynomial minimum cost flow algorithm. Oper. Res. 41, 338–350 (1993)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ramshaw, L., Tarjan, R.: Endre a weight-scaling algorithm for min-cost imperfect matchings in bipartite graphs. In: FOCS, pp. 581–590 (2012)Google Scholar
  23. 23.
    Röck, H.: Scaling techniques for minimal cost network flows. In: Pape, U. (ed.) Discrete structures and algorithms, pp. 181–191. Carl Hansen, Münich (1980)Google Scholar
  24. 24.
    Tardos, É.: Strongly polynomial minimum cost circulation algorithm. Combinatorica 5(3), 247–255 (1985)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Andrew V. Goldberg
    • 1
  • Sagi Hed
    • 2
  • Haim Kaplan
    • 2
  • Robert E. Tarjan
    • 3
    • 4
  1. 1.Amazon.com IncE. Palo AltoUSA
  2. 2.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Department of Computer SciencePrinceton UniversityPrincetonUSA
  4. 4.Intertrust TechnologiesSunnyvaleUSA

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