Integer multicommodity flows with reduced demands

  • Anand Srivastav
  • Peter Stangier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 726)


Given a supply graph G=(V, E), a demand graph H=(T, D), edge capacities c: E ↦ ℕ and requests r: D ↦ ℕ, we consider the problem of finding integer multiflows subject to c, r. Korach and Penn constructed approximate integer multiflows for planar graphs, but no results were known for the general case. Via derandomization we present a polynomial-time approximation algorithm. There are two cases:
  1. a)

    The main result is: For fractional solvable instances (G, H, c, r) and each 0 < ε ≤ 9/10 our algorithm finds in polynomial-time an integer multiflow subject to c, such that for all d ε D the d-th flow value satisfies f(d) ≥ (1-ε)r(d), provided that capacities and requests are not too small, i.e c,r = Ω(1/ε2log(¦E¦ + ¦D¦)). In particular, if c,r≥36[log 2(¦E¦+¦D¦+1)] we have a strongly polynomial-time algorithm and the first 1/2-factor approximation.

  2. b)

    If the problem is not fractionally solvable we can reduce it to the case mentioned above decreasing the requests in an optimal way. This can be done by linear programming and the results of a) apply.


The design and analysis of the algorithm require new techniques for randomized rounding as well as for derandomization. One key tool is an algorithmic version of the classical Angluin-Valiant inequality (a variant of the well known Chernoff-Hoeffding bound) estimating the tail of weighted sums of Bernoulli trials, which was not previously known and might be of independent interest in computational probability theory.

The significance of our rounding algorithm is emphasized by the fact that there is a rich combinatorial theory exhibiting many examples of fractionally solvable problems, but finding approximate integer solutions even for fractionally solvable problems is NP-hard as it is shown in this paper.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Alon, J. Spencer, P. Erdös; The probabilistic method. John Wiley & Sons, Inc. 1992.Google Scholar
  2. [2]
    D. Angluin, L.G. Valiant: Fast probabilistic algorithms for Hamiltonion circuits and matchings. J. Computer and System Sciences, Vol. 18, (1979), 155–193.Google Scholar
  3. [3]
    A. Frank: Packing paths, circuits and cuts — a survey. In Paths, Flows and VLSI-Layout, B. Korte, L. Lovasz, H.-J. Prömel, A. Schrijver (eds.), Springer Verlag, (1990), pp.47–97.Google Scholar
  4. [4]
    M. Grötschel, L. Lovász, A. Schrijver; Geometric algorithms and combinatorial optimization. Springer-Verlag (1988)Google Scholar
  5. [5]
    W. Hoeffding; On the distribution of the number of success in independent trials. Annals of Math. Stat. 27, (1956), 713–721.Google Scholar
  6. [6]
    E. Korach, M. Penn; Tight integral duality gap in the chinese postman problem Technical report, Computer Science Department, Israel Institute of Technology, Haifa, Revised Version, December 1989.Google Scholar
  7. [7]
    V. M. Malhotra, M. P. Kumar, S. N. Maheshwari; An OV¦2) Algorithm for Finding Maximum Flows in Networks. Information Processing Letters 7 (1978), 277–278.Google Scholar
  8. [8]
    C. McDiarmid; On the Method of Bounded Differences. Surveys in Combinatorics, 1989. J. Siemons, Ed.: London Math. Soc. Lectures Notes, Series 141, Cambridge University Press, Cambridge, England 1989.Google Scholar
  9. [9]
    K. Mehlhorn; Data structures and algorithms 1: Sorting and Searching. Sringer-Verlag (1984)Google Scholar
  10. [10]
    F. Pfeiffer; Personal communication, Bonn 1991.Google Scholar
  11. [11]
    P. Raghavan, C. D. Thompson; Randomized Rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (4), (1987), 365–373.Google Scholar
  12. [12]
    P. Raghavan; Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Jour. of Computer and System Sciences 37, (1988), 130–143.Google Scholar
  13. [13]
    A. Srivastav, P. Stangier; The relationship between fractional and integral graph partitioning. Working Paper; Research Institute of Discrete Mathematics, University of Bonn, (1992).Google Scholar
  14. [14]
    A. Srivastav, P. Stangier; Weighted fractional and integral k-matching in hypergraphs. Working Paper; Research Institute of Discrete Mathematics, University of Bonn (1992).Google Scholar
  15. [15]
    A. Srivastav; An Algorithmic Version of the Chernoff-Hoeffding Inequality and New Apllications to Packing Integer Programs. Working Paper; Research Institute of Discrete Mathematics, University of Bonn (1993).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Anand Srivastav
    • 1
  • Peter Stangier
    • 2
  1. 1.Research Institute of Discrete MathematicsUniversity of BonnBonnGermany
  2. 2.Institut für InformatikUniversität zu KölnKölnGermany

Personalised recommendations