Skip to main content
SpringerLink
Log in

The all-or-nothing flow problem in directed graphs with symmetric demand pairs

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF). The input consists of a directed graph \(G = (V, E)\) and a collection of (unordered) pairs of nodes \(\mathcal {M}= \left\{ s_1t_1, s_2t_2, \ldots , s_kt_k\right\} \). A subset \(\mathcal {M}'\) of the pairs is routable if there is a feasible multicommodity flow in \(G\) such that, for each pair \(s_it_i \in \mathcal {M}'\), the amount of flow from \(s_i\) to \(t_i\) is at least one and the amount of flow from \(t_i\) to \(s_i\) is at least one. The goal is to find a maximum cardinality subset of the given pairs that can be routed. Our main result is a poly-logarithmic approximation with constant congestion for SymANF. We obtain this result by extending the well-linked decomposition framework of Chekuri et al. (2005) to the directed graph setting with symmetric demand pairs. We point out the importance of studying routing problems in this setting and the relevance of our result to future work.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. A routing has congestion \(c\) if it violates the capacities by a factor of at most \(c\).

  2. There are alternative ways to define routability that captures symmetry. One option is to require a flow of \(1/2\) unit in each direction which is compatible with a total of one unit of flow entering and leaving each terminal. Another option is to require that for any orientation of the demand pairs, there is a feasible multicommodity for the pairs with one unit for each pair in the direction given by the orientation; however, deciding the routability according to this definition is not easy. For simplicity we require one unit of flow in each direction which results in a factor of \(2\) loss in the congestion when compared to other models.

  3. The \(\tilde{\varOmega }\) notation hides poly-logarithmic factors.

  4. The implications of crossbar results for product multicommodity flow-cut gaps is pointed out in [9].

  5. The symANF-LP relaxation is given in Sect. 3.3.

References

  1. Adler, I.: Directed tree-width examples. J Comb. Theory Ser. B 97(5), 718–725 (2007)

    Article  MATH  Google Scholar 

  2. Andrews, M., Chuzhoy, J., Guruswami, V., Khanna, S., Talwar, K., Zhang, L.: Inapproximability of edge-disjoint paths and low congestion routing on undirected graphs. Combinatorica 30(5), 485–520 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andrews, M., Chuzhoy, J., Khanna, S., Zhang, L.: Hardness of the undirected edge-disjoint paths problem with congestion. In: Proceeding of IEEE FOCS, pp. 226–241 (2005)

  4. Chekuri, C., Chuzhoy, J.: Large-treewidth graph decompositions and applications. In: Proceedings of ACM STOC (2013)

  5. Chekuri, C., Chuzhoy, J.: Polynomial bounds for the grid-minor theorem. In: Proceedings of ACM STOC (2014)

  6. Chekuri, C., Ene, A.: Poly-logarithmic approximation for maximum node disjoint paths with constant congestion. In: Proceedings of ACM-SIAM SODA (2013)

  7. Chekuri, C., Kannan, S., Raja, A., Viswanath, P.: Multicommodity flows and cuts in polymatroidal networks. In: Proceedings of ITCS, pp. 399–408 (2012)

  8. Chekuri, C., Khanna, S., Shepherd, F.B.: The all-or-nothing multicommodity flow problem. SIAM J. Comput. 42(4), 1467–1493 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chekuri, C., Khanna, S., Shepherd, F.B.: Multicommodity flow, well-linked terminals, and routing problems. In: Proceedings of ACM STOC, pp. 183–192 (2005)

  10. Chekuri, C., Khanna, S., Shepherd, F.B.: Well-linked terminals for node-capacitated routing problems. Manuscript (2005)

  11. Chekuri, C., Khanna, S., Shepherd, F.B.: An \(O(\sqrt{n})\) approximation and integrality gap for disjoint paths and unsplittable flow. Theory Comput. 2(7), 137–146 (2006)

    Article  MathSciNet  Google Scholar 

  12. Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity demand flow in a tree and packing integer programs. ACM Trans. Algorithms 3(3), 27 (2007)

    Article  MathSciNet  Google Scholar 

  13. Chuzhoy, J.: Routing in undirected graphs with constant congestion. ArXiv preprint arXiv:1107.2554 (2011). Extended abstract in STOC 2012

  14. Chuzhoy, J., Guruswami, V., Khanna, S., Talwar, K.: Hardness of routing with congestion in directed graphs. In: Proceedings of ACM STOC, pp. 165–178 (2007)

  15. Chuzhoy, J., Khanna, S.: Polynomial flow-cut gaps and hardness of directed cut problems. J. ACM 56(2), 6 (2009)

    Article  MathSciNet  Google Scholar 

  16. Chuzhoy, J., Li, S.: A polylogarithimic approximation algorithm for edge-disjoint paths with congestion 2. In: Proceedings of IEEE FOCS (2012)

  17. Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM J. Comput. 5(4), 691–703 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  18. Feige, U., Hajiaghayi, M.T., Lee, J.R.: Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput. 38, 629–657 (2008)

    Article  MathSciNet  Google Scholar 

  19. Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10(2), 111–121 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  20. Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18(1), 3–20 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. J. Comb. Theory Ser. B 82(1), 138–154 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum press, New York (1972)

  23. Kawarabayashi, K., Kreutzer, S.: An excluded grid theorem for digraphs with forbidden minors. In: Proceedings of ACM-SIAM SODA (2014)

  24. Kobayashi, Y., Kawarabayashi, K-I, Kreutzer, S.: An excluded half-integral grid theorem for digraphs and the directed disjoint paths problem. In: Proceedings of ACM STOC (2014)

  25. Klein, P.N., Plotkin, S.A., Rao, S.: Excluded minors, network decomposition, and multicommodity flow. In: Proceedings of ACM STOC, pp. 682–690 (1993)

  26. Klein, P.N., Plotkin, S.A., Rao, S., Tardos, E.: Approximation algorithms for Steiner and directed multicuts. J. Algorithms 22(2), 241–269 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46(6), 787–832 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–245 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  29. Reed, B.: Introducing directed tree width. Electr. Notes Discr. Math. 3, 222–229 (1999)

    Article  Google Scholar 

  30. Saks, Michael E., Samorodnitsky, Alex, Zosin, Leonid: A lower bound on the integrality gap for minimum multicut in directed networks. Combinatorica 24(3), 525–530 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Srinivasan, A.: Improved approximations for edge-disjoint paths, unsplittable flow, and related routing problems. In: Proceedings of IEEE FOCS, pp. 416–425 (1997)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Chicago, IL, USA

    Chandra Chekuri

  2. Center for Computational Intractability, Princeton University, Princeton, NJ, USA

    Alina Ene

  3. Department of Computer Science and DIMAP, University of Warwick, Coventry, UK

    Alina Ene

Authors
  1. Chandra Chekuri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alina Ene
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alina Ene.

Additional information

Chandra Chekuri: supported in part by NSF Grants CCF-1016684 and CCF-1319376. Part of this work was done while the author was supported by TTI Chicago on a sabbatical visit in Fall 2012. Alina Ene: supported in part by NSF Grants CCF-1016684 and CCF-0844872. Part of this work was done while the author was an intern at TTI Chicago.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chekuri, C., Ene, A. The all-or-nothing flow problem in directed graphs with symmetric demand pairs. Math. Program. 154, 249–272 (2015). https://doi.org/10.1007/s10107-014-0856-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-014-0856-z

Mathematics Subject Classification

Search

Navigation