Skip to main content
Springer Nature Link
Log in

Improved Algorithms for Even Factors and Square-Free Simple b-Matchings

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Given a digraph G=(VG,AG), an even factor M⊆AG is a set formed by node-disjoint paths and even cycles. Even factors in digraphs were introduced by Geelen and Cunningham and generalize path matchings in undirected graphs.

Finding an even factor of maximum cardinality in a general digraph is known to be NP-hard but for the class of odd-cycle symmetric digraphs the problem is polynomially solvable. So far the only combinatorial algorithm known for this task is due to Pap; its running time is O(n 4) (hereinafter n denotes the number of nodes in G and m denotes the number of arcs or edges).

In this paper we introduce a novel sparse recovery technique and devise an O(n 3logn)-time algorithm for finding a maximum cardinality even factor in an odd-cycle symmetric digraph. Our technique also applies to other similar problems, e.g. finding a maximum cardinality square-free simple b-matching.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2
Algorithm 2
Fig. 3
Fig. 4
Algorithm 3

Similar content being viewed by others

References

  1. Babenko, M.: A faster algorithm for the maximum even factor problem. In: Proc. 21st International Symposium on Algorithms and Computation, pp. 451–462 (2010)

    Google Scholar 

  2. Balas, E., Pulleyblank, W.: The perfectly matchable subgraph polytope of an arbitrary graph. Combinatorica 9, 321–337 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cormen, T., Stein, C., Rivest, R., Leiserson, C.: Introduction to Algorithms. McGraw-Hill Higher Education, Boston (2001)

    MATH  Google Scholar 

  4. Cunningham, W.H.: Matching, matroids, and extensions. Math. Program. 91(3), 515–542 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cunningham, W.H., Geelen, J.F.: The optimal path-matching problem. Combinatorica 17, 315–337 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cunningham, W.H., Geelen, J.F.: Combinatorial algorithms for path-matching. Manuscript (2000)

  7. Cunningham, W.H., Geelen, J.F.: Vertex-disjoint dipaths and even dicircuits. Manuscript (2001)

  8. Hartvigsen, D.: Finding maximum square-free 2-matchings in bipartite graphs. J. Comb. Theory, Ser. B

  9. Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Iwata, S., Takazawa, K.: The independent even factor problem. SIAM J. Discrete Math. 22, 1411–1427 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Király, Z.: C 4-free 2-factors in bipartite graphs. EGRES Technical Report TR-2001-13 (2001)

  12. Kobayashi, Y., Takazawa, K.: Even factors, jump systems, and discrete convexity. J. Comb. Theory, Ser. B 99(1), 139–161 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Micali, S., Vazirani, V.: An \(O(\sqrt{| V |} \cdot | E |)\) algorithm for finding maximum matching in general graphs. In: Proc. 45st IEEE Symp. Foundations of Computer Science, pp. 248–255 (1980)

    Google Scholar 

  14. Pap, G.: A combinatorial algorithm to find a maximum even factor. In: Proc. 11th Integer International IPCO Conference on Programming and Combinatorial Optimization, pp. 66–80 (2005)

    Google Scholar 

  15. Pap, G.: Combinatorial algorithms for matchings, even factors and square-free 2-factors. Math. Program. 110(1), 57–69 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pap, G., Szegö, L.: On the maximum even factor in weakly symmetric graphs. J. Comb. Theory, Ser. B 91(2), 201–213 (2004)

    Article  MATH  Google Scholar 

  17. Spille, B., Weismantel, R.: A generalization of Edmonds’ matching and matroid intersection algorithms. In: Proc. 9th International IPCO Conference on Integer Programming and Combinatorial Optimization, pp. 9–20 (2002)

    Google Scholar 

  18. Takazawa, K.: A weighted even factor algorithm. Math. Program., Ser. A, B 115(2), 223–237 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tarjan, R.: Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics, Philadelphia (1983)

    Book  Google Scholar 

Download references

Acknowledgement

The author is thankful to the anonymous referees for useful comments and suggestions.

Author information

Authors and Affiliations

  1. Dept. of Mechanics and Mathematics, Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russia

    Maxim A. Babenko

  2. Yandex LLC, 16, Leo Tolstoy St., Moscow, 119021, Russia

    Maxim A. Babenko

Authors
  1. Maxim A. Babenko
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Maxim A. Babenko.

Additional information

Supported by RFBR grant 09-01-00709-a.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Babenko, M.A. Improved Algorithms for Even Factors and Square-Free Simple b-Matchings. Algorithmica 64, 362–383 (2012). https://doi.org/10.1007/s00453-012-9642-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-012-9642-6

Keywords