Skip to main content
SpringerLink
Log in

An Algorithm for the Polyhedral Cycle Cover Problem with Constraints on the Number and Length of Cycles

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

A cycle cover of a graph is a spanning subgraph whose connected components are simple cycles. Given a complete weighted directed graph, consider the intractable problem of finding a maximum-weight cycle cover which satisfies an upper bound on the number of cycles and a lower bound on the number of edges in each cycle. We suggest a polynomial-time algorithm for solving this problem in the geometric case where the vertices of the graph are points in a multidimensional real space and the distances between them are induced by a positively homogeneous function whose unit ball is an arbitrary convex polytope with a fixed number of facets. The obtained result extends the ideas underlying the well-known algorithm for the polyhedral Max TSP.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. I. Serdyukov, “An asymptotically optimal algorithm for the maximum traveling salesman problem in Euclidean space,” in Integer Optimization Methods. Controlled Systems (SO AN SSSR, Novosibirsk, 1987), issue 27, pp. 79–87 [in Russian].

    Google Scholar 

  2. A. I. Serdyukov, “The maximum-weight traveling salesman problem in finite-dimensional real spaces,” in Operations Research and Discrete Analysis (Kluwer, Dordrecht, 1997), Ser. Mathematics and Its Applications 391, pp. 233–239.

    Chapter  Google Scholar 

  3. V. V. Shenmaier, “An asymptotically exact algorithm for the maximum traveling salesman problem in a finite-dimensional normed space,” J. Appl. Ind. Math. 5 (2), 296–300 (2011).

    Article  MathSciNet  Google Scholar 

  4. A. Barvinok, S. P. Fekete, D. S. Johnson, A. Tamir, G. J. Woeginger, and R. Woodroofe, “The geometric maximum traveling salesman problem,” J. ACM 50 (5), 641–664 (2003). doi https://doi.org/10.1145/876638.876640

    Article  MathSciNet  Google Scholar 

  5. M. Bläser and B. Manthey, “Approximating maximum weight cycle covers in directed graphs with weights zero and one,” Algorithmica 42 (2), 121–139 (2005). doi https://doi.org/10.1007/s00453-004-1131-0

    Article  MathSciNet  Google Scholar 

  6. A. Cayley, “A theorem on trees,” Quart. J. Pure Appl. Math. 23, 376–378 (1889).

    MATH  Google Scholar 

  7. L. Engebretsen and M. Karpinski, “TSP with bounded metrics,” J. Comp. System Sci. 72 (4), 509–546 (2006). doi https://doi.org/10.1016/j.jcss.2005.12.001

    Article  MathSciNet  Google Scholar 

  8. H. Kaplan, M. Lewenstein, N. Shafrir, and M. Sviridenko, “Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs,” J. ACM 52 (4), 602–626 (2005). doi https://doi.org/10.1145/1082036.1082041

    Article  MathSciNet  Google Scholar 

  9. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Ed. by E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. Shmoys (Wiley, Chichester, 1985).

    MATH  Google Scholar 

  10. B. Manthey, “On approximating restricted cycle covers,” in Proceedings of the 3rd Workshop on Approximation and Online Algorithms, Palma de Mallorca, Spain, 2005 (Springer, Berlin, 2006), Ser. Lecture Notes in Computer Science 3879, pp. 282–295.

    Google Scholar 

  11. K. Paluch, M. Mucha, and A. Madry, “A 7/9-approximation algorithm for the maximum traveling salesman problem,” in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (Springer, Berlin, 2009), Ser. Lecture Notes in Computer Science 5687, pp. 298–311.

    Chapter  Google Scholar 

  12. V. V. Shenmaier, “Asymptotically optimal algorithms for geometric Max TSP and Max m-PSP,” Discrete Appl. Math. 163, 214–219 (2014). doi https://doi.org/10.1016/j.dam.2012.09.007

    Article  MathSciNet  Google Scholar 

  13. V. V. Shenmaier, “Complexity and approximation of the smallest k-enclosing ball problem,” European J. Combin. 48, 81–87 (2015). doi https://doi.org/10.1016/j.ejc.2015.02.011

    Article  MathSciNet  Google Scholar 

  14. E. Zemel, “An O(n) algorithm for the linear multiple choice knapsack problem and related problems,” Inform. Process. Lett. 18 (3), 123–128 (1984). doi https://doi.org/10.1016/0020-0190(84)90014-0

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation (project no. 16-11-10041).

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. V. Shenmaier

Authors
  1. V. V. Shenmaier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Shenmaier.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shenmaier, V.V. An Algorithm for the Polyhedral Cycle Cover Problem with Constraints on the Number and Length of Cycles. Proc. Steklov Inst. Math. 307 (Suppl 1), 142–150 (2019). https://doi.org/10.1134/S0081543819070113

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543819070113

Keywords

Search

Navigation