Advertisement

The All-or-Nothing Flow Problem in Directed Graphs with Symmetric Demand Pairs

  • Chandra Chekuri
  • Alina Ene
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8494)

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 \({\mathcal{M}} = \{s_1t_1, s_2t_2, \dots, s_kt_k\}\). 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 [6] 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andrews, M., Chuzhoy, J., Khanna, S., Zhang, L.: Hardness of the undirected edge-disjoint paths problem with congestion. In: Proc. of IEEE FOCS, pp. 226–241 (2005)Google Scholar
  3. 3.
    Chekuri, C., Chuzhoy, J.: Large-treewidth graph decompositions and applications. In: Proc. of ACM STOC (2013)Google Scholar
  4. 4.
    Chekuri, C., Chuzhoy, J.: Polynomial bounds for the grid-minor theorem. In: Proc. of ACM STOC (2014)Google Scholar
  5. 5.
    Chekuri, C., Khanna, S., Shepherd, F.B.: The all-or-nothing multicommodity flow problem. SIAM Journal on Computing 42(4), 1467–1493 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chekuri, C., Khanna, S., Shepherd, F.B.: Multicommodity flow, well-linked terminals, and routing problems. In: Proc. of ACM STOC, pp. 183–192 (2005)Google Scholar
  7. 7.
    Chekuri, C., Khanna, S., Shepherd, F.B.: Well-linked terminals for node-capacitated routing problems (2005) (Manuscript)Google Scholar
  8. 8.
    Chekuri, C., Khanna, S., Shepherd, F.B.: An \(O(\sqrt{n})\) approximation and integrality gap for disjoint paths and unsplittable flow. Theory of Computing 2(7), 137–146 (2006)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chekuri, C., Mydlarz, M., Shepherd, F.B.: Multicommodity demand flow in a tree and packing integer programs. ACM Transactions on Algorithms 3(3), 27 (2007)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Chuzhoy, J., Guruswami, V., Khanna, S., Talwar, K.: Hardness of routing with congestion in directed graphs. In: Proc. of ACM STOC, pp. 165–178 (2007)Google Scholar
  11. 11.
    Chuzhoy, J., Li, S.: A polylogarithimic approximation algorithm for edge-disjoint paths with congestion 2. In: Proc. of IEEE FOCS (2012)Google Scholar
  12. 12.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory, Series B 82(1), 138–154 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Reed, B.: Introducing directed tree width. Electronic Notes in Discrete Mathematics 3, 222–229 (1999)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Chandra Chekuri
    • 1
  • Alina Ene
    • 2
    • 3
  1. 1.Dept. of Computer ScienceUniversity of Illinois at Urbana-ChampaignUSA
  2. 2.Center of Computational IntractabilityPrinceton UniversityUSA
  3. 3.Dept. of Computer Science and DIMAPUniversity of WarwickUK

Personalised recommendations