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Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

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Abstract

An elementary h-route flow, for an integer h≥1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h≤3: The size of a minimum h-route cut is at least f/h and at most O(log4 kf) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h=3 that has an approximation ratio of O(log4 k). Previously, polylogarithmic approximation was known only for h-route cuts for h≤2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.

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Notes

  1. To be precise, the entry edges of H(r) are a subset of δ(r)∖δ 2(r), and the entry nodes of H(r) are a subset of Z: it may happen that (some of) the nodes in Z are not connected to an outside node (see the definition of entry nodes). However, such a pathological case makes the situation only easier and thus, without loss of generality, we assume that all nodes in Z are the entry nodes of H(r).

References

  1. Aggarwal, C.C., Orlin, J.B.: On multiroute maximum flows in networks. Networks 39, 43–52 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aumann, Y., Rabani, Y.: An O(logk) approximate min-cut max-flow theorem and approximation algorithm. SIAM J. Comput. 27(1), 291–301 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bagchi, A., Chaudhary, A., Kolman, P., Sgall, J.: A simple combinatorial proof for the duality of multiroute flows and cuts. Technical Report 2004-662, Charles University, Prague (2004)

  4. Baier, G., Erlebach, T., Hall, A., Köhler, E., Kolman, P., Pangrác, O., Schilling, H., Skutella, M.: Length-bounded cuts and flows. ACM Trans. Algorithms 7(1), 4–27 (2010)

    Article  MathSciNet  Google Scholar 

  5. Barman, S., Chawla, S.: Region growing for multi-route cuts. In: Proceedings of the 21th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2010)

    Google Scholar 

  6. Bruhn, H., Černý, J., Hall, A., Kolman, P., Sgall, J.: Single source multiroute flows and cuts on uniform capacity networks. Theory Comput. 4(1), 1–20 (2008). Preliminary version in Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2007)

    Article  MathSciNet  Google Scholar 

  7. Chekuri, C., Khanna, S.: Algorithms for 2-route cut problems. In: Proceedings of the 35th International Colloquium on Automata (ICALP). Lecture Notes in Computer Science, vol. 5125, pp. 472–484 (2008)

    Google Scholar 

  8. Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23(4), 864–894 (1994). Preliminary version in Proceedings of the 24th ACM Symposium on Theory of Computing (STOC) (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ford, L.R., Fulkerson, D.R.: Maximum flow through a network. Can. J. Math. 8, 399–404 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  10. Garg, N., Vazirani, V.V., Yannakakis, M.: Approximate max-flow min-cut theorems and their applications. SIAM J. Comput. 25(2), 235–251 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kahale, N.: On reducing the cut ratio to the multicut problem. DIMACS Technical report 93-78 (1993)

  12. Kishimoto, W., Takeuchi, M.: On m-route flows in a network. IEICE Trans. J-76-A(8), 1185–1200 (1993) (in Japanese)

    Google Scholar 

  13. Kolman, P., Scheideler, C.: Towards duality of multicommodity multiroute cuts and flows: multilevel ball-growing. In: Proceedings of the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), Leibniz International Proceedings in Informatics (LIPIcs) (2011)

    Google Scholar 

  14. Kolman, P., Scheideler, C.: Approximate duality of multicommodity multiroute flows and cuts: single source case. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 800–810 (2012)

    Google Scholar 

  15. Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46(6), 787–832 (1999). Preliminary version in Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 215–245 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shmoys, D.B.: Cut problems and their application to divide-and-conquer. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-hard Problems, pp. 192–235. PWS, Boston (1997)

    Google Scholar 

  18. Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2001)

    Google Scholar 

  19. Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

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Acknowledgements

The first author would like to thank Jiří Sgall and Thomas Erlebach for stimulating discussions.

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Correspondence to Christian Scheideler.

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P. Kolman was supported by the Center of Excellence—Institute for Theoretical Computer Science, Prague, project P202/12/G061 of GA ČR. C. Scheideler was supported by DFG SCHE 1592/1-1.

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Kolman, P., Scheideler, C. Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing. Theory Comput Syst 53, 341–363 (2013). https://doi.org/10.1007/s00224-013-9454-3

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