Mathematical Methods of Operations Research

, Volume 77, Issue 3, pp 381–391 | Cite as

A polynomial time approximation algorithm for the two-commodity splittable flow problem

Original Article

Abstract

We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity \({i \in \{1, 2\}}\) can be split into a bounded number ki of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even ki and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k1, k2)-splittable flow without chunk size restrictions for fixed demand ratios.

Keywords

Splittable flow 2-commodity flow Approximation algorithm 

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References

  1. Baier G, Köhler E, Skutella M (2002) On the k-splittable flow problem. In: Algorithms—ESA 2002. Lecture notes in computer science, vol. 2461. Springer, Berlin, pp 101–113. doi:10.1007/3-540-45749-6_13
  2. Baier G, Köhler E, Skutella M (2005) The k-splittable flow problem. Algorithmica 42: 231–248. doi:10.1007/s00453-005-1167-9 MathSciNetMATHCrossRefGoogle Scholar
  3. Even S, Itai A, Shamir A (1976) On the complexity of timetable and multicommodity flow problems. SIAM J Comput 5(4): 691–703. doi:10.1137/0205048 MathSciNetMATHCrossRefGoogle Scholar
  4. Hu TC (1963) Multi-commodity network flows. Oper Res 11(3): 344–360. doi:10.1287/opre.11.3.344 MATHCrossRefGoogle Scholar
  5. Kleinberg JM (1996) Approximation algorithms for disjoint paths problems. Ph.d. thesis, Massachusetts Institute of TechnologyGoogle Scholar
  6. Schrijver A (2003) Combinatorial optimization. Polyhedra and efficiency. Vol. C, Algorithms and Combinatorics, vol. 24. Springer, BerlinGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institut für Mathematische OptimierungOtto-von-Guericke Universität MagdeburgMagdeburgGermany
  2. 2.Institut für Operations ResearchETH ZürichZurichSwitzerland

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