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Maximum Flow in Directed Planar Graphs with Vertex Capacities

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Abstract

In this paper we present an O(nlog n) time algorithm for finding a maximum flow in a directed planar graph, where the vertices are subject to capacity constraints, in addition to the arcs. If the source and the sink are on the same face, then our algorithm can be implemented in O(n) time.

For general (not planar) graphs, vertex capacities do not make the maximum flow problem more difficult, as there is a simple reduction that eliminates vertex capacities. However, this reduction does not preserve the planarity of the graph. The essence of our algorithm is a different reduction that does preserve the planarity and can be implemented in linear time. For the special case of undirected planar graphs, an algorithm with the same time complexity was recently claimed, but we show that it has a flaw.

We also apply our technique to obtain a linear-time algorithm to convert a flow to an acyclic flow, and a linear-time algorithm to find a largest set of vertex-disjoint s−t paths, in a directed planar graph.

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References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  2. Borradaile, G., Klein, P.: An O(nlog n) algorithm for maximum st-flow in a directed planar graph. J. ACM 56, 1–30 (2009)

    MathSciNet  Google Scholar 

  3. Brandes, U., Wagner, D.: A linear time algorithm for the arc disjoint Menger problem in planar directed graphs. Algorithmica 28, 16–36 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton (1962)

    MATH  Google Scholar 

  5. Hassin, R.: Maximum flow in (s,t) planar networks. Inf. Process. Lett. 13, 107 (1981)

    Article  MathSciNet  Google Scholar 

  6. Hassin, R., Johnson, D.B.: An O(nlog 2 n) algorithm for maximum flow in undirected planar networks. SIAM J. Comput. 14, 612–624 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  7. Henzinger, M.R., Klein, P., Rao, S., Subramania, S.: Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci. 55, 3–23 (1997)

    Article  MATH  Google Scholar 

  8. Itai, A., Shiloach, Y.: Maximum flow in planar networks. SIAM J. Comput. 8, 135–150 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Johnson, D.B.: Parallel algorithms for minimum cuts and maximum flows in planar networks. J. ACM 34, 950–967 (1987)

    Article  Google Scholar 

  10. Khuller, S., Naor, J.: Flow in planar graphs with vertex capacities. Algorithmica 11, 200–225 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Khuller, S., Naor, J., Klein, P.: The lattice structure of flow in planar graphs. SIAM J. Discrete Math. 63, 477–490 (1993)

    Article  MathSciNet  Google Scholar 

  12. Nishizwki, T., Chiba, N.: Planar Graphs: Theory and Algorithms. Ann. Discrete Math., vol. 32. North-Holland, Amsterdam (1988)

    Google Scholar 

  13. Ripphausen-Lipa, H., Wagner, D., Weihe, K.: The vertex-disjoint Menger problem in planar graphs. SIAM J. Comput. 26, 331–349 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26, 362–391 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  15. Weihe, K.: Edge-disjoint (s,t)-paths in undirected planar graphs in linear time. J. Algorithms 23, 121–138 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Weihe, K.: Maximum (s,t)-flows in planar networks in O(|V|log |V|)-time. J. Comput. Syst. Sci. 55, 454–476 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, X., Liang, W., Jiang, H.: Flow equivalent trees in node-edge-capacitated undirected planar graphs. Inf. Process. Lett. 100, 100–115 (2006)

    Article  MathSciNet  Google Scholar 

  18. Zhang, X., Liang, W., Chen, G.: Computing maximum flows in undirected planar networks with both edge and vertex capacities. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 577–586. Springer, Heidelberg (2008)

    Google Scholar 

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Authors and Affiliations

  1. The Blavatnik School of Computer Science, Tel Aviv University, 69978, Tel Aviv, Israel

    Haim Kaplan & Yahav Nussbaum

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  1. Haim Kaplan
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  2. Yahav Nussbaum
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Correspondence to Yahav Nussbaum.

Additional information

This research was partially supported by the United States—Israel Binational Science Foundation, project number 2006204.

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Kaplan, H., Nussbaum, Y. Maximum Flow in Directed Planar Graphs with Vertex Capacities. Algorithmica 61, 174–189 (2011). https://doi.org/10.1007/s00453-010-9436-7

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  • DOI: https://doi.org/10.1007/s00453-010-9436-7

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