Advertisement

Packing Directed Cycles Efficiently

  • Zeev Nutov
  • Raphael Yuster
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3153)

Abstract

Let G be a simple digraph. The dicycle packing number of G, denoted ν c (G), is the maximum size of a set of arc-disjoint directed cycles in G. Let G be a digraph with a nonnegative arc-weight function w. A function ψ from the set \({\cal C}\) of directed cycles in G to R  +  is a fractional dicycle packing of G if \(\sum_{e \in C \in {\cal C}} {\psi(C)} \leq w(e)\) for each eE(G). The fractional dicycle packing number, denoted ν \(_{c}^{\rm *}\)(G,w), is the maximum value of \(\sum_{C \in {\cal C}} \psi(C)\) taken over all fractional dicycle packings ψ. In case w ≡ 1 we denote the latter parameter by ν \(_{c}^{\rm *}\)(G).

Our main result is that ν \(_{c}^{\rm *}\)(G) – ν c (G)=o(n 2) where n=|V(G)|. Our proof is algorithmic and generates a set of arc-disjoint directed cycles whose size is at least ν c (G)-o(n 2) in randomized polynomial time. Since computing ν c (G) is an NP-Hard problem, and since almost all digraphs have ν c (G)=Θ(n 2) our result is a FPTAS for computing ν c (G) for almost all digraphs.

The latter result uses as its main lemma a much more general result. Let \({\cal F}\) be any fixed family of oriented graphs. For an oriented graph G, let \(\nu_{\cal F}(G)\) denote the maximum number of arc-disjoint copies of elements of \({\cal F}\) that can be found in G, and let \(\nu_{\cal F}^*(G)\) denote the fractional relaxation. Then, \(\nu_{\cal F}^*(G) - \nu_{\cal F}(G)=o(n^2)\). This lemma uses the recently discovered directed regularity lemma as its main tool.

It is well known that ν \(_{c}^{\rm *}\)(G,w) can be computed in polynomial time by considering the dual problem. However, it was an open problem whether an optimal fractional dicycle packing ψyielding ν \(_{c}^{\rm *}\)(G,w) can be generated in polynomial time. We prove that a maximum fractional dicycle packing yielding ν \(_{c}^{\rm *}\)(G,w) with at most O(n 2) dicycles receiving nonzero weight can be found in polynomial time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Duke, R.A., Lefmann, H., Rödl, V., Yuster, R.: The algorithmic aspects of the Regularity Lemma. Journal of Algorithms 16, 80–109 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Shapira, A.: Testing subgraphs in directed graphs. In: Proc. 35th ACM STOC, pp. 700–709. ACM Press, New York (2003)Google Scholar
  3. 3.
    Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. Wiley, New York (2000)zbMATHCrossRefGoogle Scholar
  4. 4.
    Balister, P.: Packing digraphs with directed closed trails. Combin. Probab. Comput. 12, 1–15 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bollobás, B.: Extremal Graph Theory. Academic Press, London (1978)zbMATHGoogle Scholar
  6. 6.
    Dor, D., Tarsi, M.: Graph decomposition is NPC - A complete proof of Holyer’s conjecture. In: Proc. 20th ACM STOC, pp. 252–263. ACM Press, New York (1992)Google Scholar
  7. 7.
    Frankl, P., Rödl, V.: Near perfect coverings in graphs and hypergraphs. European J. Combinatorics 6, 317–326 (1985)zbMATHGoogle Scholar
  8. 8.
    Füredi, Z.: Matchings and covers in hypergraphs. Graphs and Combinatorics 4, 115–206 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grable, D.: Nearly-perfect hypergraph packing is in NC. Information Processing Letters 60, 295–299 (1996)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)zbMATHGoogle Scholar
  11. 11.
    Haxell, P.E., Rödl, V.: Integer and fractional packings in dense graphs. Combinatorica 21, 13–38 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Krivelevich, M., Nutov, Z., Yuster, R.: Approximation algorithms for cycle packing problems (preprint)Google Scholar
  13. 13.
    Nutov, Z., Penn, M.: On the integral dicycle packings and covers and the linear ordering polytope. Discrete Applied Math. 60, 293–309 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Seymour, P.D.: Packing directed circuits fractionally. Combinatorica 15, 281–288 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Szemerédi, E.: Regular partitions of graphs. In: Proc. Colloque Inter. CNRS 260, CNRS, Paris, pp. 399–401 (1978)Google Scholar
  16. 16.
    Tardos, É.: A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research 34, 250–256 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Yuster, R.: Integer and fractional packing of families of graphs, Random Structures and Algorithms (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Zeev Nutov
    • 1
  • Raphael Yuster
    • 2
  1. 1.Department of Computer ScienceThe Open University of IsraelTel AvivIsrael
  2. 2.Department of MathematicsUniversity of Haifa at OranimTivonIsrael

Personalised recommendations