Approximation Algorithms for Restricted Cycle Covers Based on Cycle Decompositions

  • Bodo Manthey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4271)


A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L ⊆ ℕ. For most sets L, the problem of computing L-cycle covers of maximum weight is NP-hard and APX-hard.

We devise polynomial-time approximation algorithms for L-cycle covers. More precisely, we present a factor 2 approximation algorithm for computing L-cycle covers of maximum weight in undirected graphs and a factor 20/7 approximation algorithm for the same problem in directed graphs. Both algorithms work for arbitrary sets L. To do this, we develop a general decomposition technique for cycle covers.

Finally, we show tight lower bounds for the approximation ratios achievable by algorithms based on such decomposition techniques.


Approximation Algorithm Directed Graph Undirected Graph Approximation Ratio Maximum Weight 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  3. 3.
    Bläser, M.: A 3/4-approximation algorithm for maximum ATSP with weights zero and one. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 61–71. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Bläser, M., Manthey, B.: Approximating maximum weight cycle covers in directed graphs with weights zero and one. Algorithmica 42(2), 121–139 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bläser, M., Manthey, B., Sgall, J.: An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality. J. Discrete Algorithms (to appear)Google Scholar
  6. 6.
    Bläser, M., Shankar Ram, L., Sviridenko, M.I.: Improved approximation algorithms for metric maximum ATSP and maximum 3-cycle cover problems. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 350–359. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Bläser, M., Siebert, B.: Computing cycle covers without short cycles. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 368–379. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Blum, A.L., Jiang, T., Li, M., Tromp, J., Yannakakis, M.: Linear approximation of shortest superstrings. J. ACM 41(4), 630–647 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S., Unger, W.: Approximation algorithms for the TSP with sharpened triangle inequality. Inform. Process. Lett. 75(3), 133–138 (2000)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Sunil Chandran, L., Shankar Ram, L.: Approximations for ATSP with parametrized triangle inequality. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 227–237. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Chen, Z.-Z., Nagoya, T.: Improved approximation algorithms for metric Max TSP. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 179–190. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Chen, Z.-Z., Okamoto, Y., Wang, L.: Improved deterministic approximation algorithms for Max TSP. Inform. Process. Lett. 95(2), 333–342 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hartvigsen, D.: An Extension of Matching Theory. PhD thesis, Carnegie Mellon University, Pittsburgh, USA (1984)Google Scholar
  14. 14.
    Hassin, R., Rubinstein, S.: On the complexity of the k-customer vehicle routing problem. Oper. Res. Lett. 33(1), 71–76 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hassin, R., Rubinstein, S.: An approximation algorithm for maximum triangle packing. Discrete Appl. Math. 154(6), 971–979 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hell, P., Kirkpatrick, D.G., Kratochvíl, J., Kríz, I.: On restricted two-factors. SIAM J. Discrete Math. 1(4), 472–484 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.: Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. J. ACM 52(4), 602–626 (2005)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lovász, L., Plummer, M.D.: Matching Theory. In: North-Holland Mathematics Studies, vol. 121. Elsevier, Amsterdam (1986)Google Scholar
  19. 19.
    Manthey, B.: On approximating restricted cycle covers. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 282–295. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. In: Algorithms and Combinatorics, vol. 24. Springer, Heidelberg (2003)Google Scholar
  21. 21.
    Sweedyk, Z.: A \(2\frac12\)-approximation algorithm for shortest superstring. SIAM J. Comput. 29(3), 954–986 (1999)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Vornberger, O.: Easy and hard cycle covers. Technical report, Universität/Gesamthochschule Paderborn (1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bodo Manthey
    • 1
  1. 1.InformatikUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations