Skip to main content

Counting Cycles on Planar Graphs in Subexponential Time

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13595))

Included in the following conference series:

  • 746 Accesses

Abstract

We study the problem of counting all cycles or self-avoiding walks (SAWs) on triangulated planar graphs. We present a subexponential \(2^{O(\sqrt{n})}\) time algorithm for this counting problem. Among the technical ingredients used in this algorithm are the planar separator theorem and a delicate analysis using pairs of Motzkin paths and Motzkin numbers. We can then adapt this algorithm to uniformly sample SAWs, in subexponential time. Our work is motivated by the problem of gerrymandered districting maps.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The algorithm in [4] is applicable for grid graphs only, and it was explicitly calculated that the number of self avoiding walks connecting two diagonal corners in a \(19 \times 19\) grid graph is \( > 10^{88}\). Our algorithm for planar graphs is based on a recursive, thus different, approach.

  2. 2.

    It is known that asymptotically we have \( \left|\mathcal {L}_{A} \right|\le O(3^{|E_{A}|})\).

References

  1. Aigner, M.: Motzkin numbers. Eur. J. Comb. 19(6), 663–675 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Seymour, P., Thomas, R.: Planar separators. SIAM J. Discrete Math. 7(2), 184–193 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bousquet-Mélou, M., Guttmann, A.J., Jensen, I.: Self-avoiding walks crossing a square. J. Phys. A: Math. Gen. 38(42), 9159 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, J.Y., Maran, A.: Counting cycles on planar graphs in subexponential time (2022). https://doi.org/10.48550/ARXIV.2208.09948, https://arxiv.org/abs/2208.09948

  6. Djidjev, H.N., Venkatesan, S.M.: Reduced constants for simple cycle graph separation. Acta Informatica 34(3), 231–243 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.V.: Efficient exact algorithms on planar graphs: exploiting sphere cut decompositions. Algorithmica 58(3), 790–810 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Har-Peled, S.: A simple proof of the existence of a planar separator. arXiv preprint arXiv:1105.0103 (2011)

  9. Jerrum, M.R., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43, 169–188 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Johnson, D.B.: Finding all the elementary circuits of a directed graph. SIAM J. Comput. 4(1), 77–84 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Miller, G.L.: Finding small simple cycle separators for 2-connected planar graphs. J. Comput. Syst. Sci. 32(3), 265–279 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Motzkin, T.: Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products. Bull. Am. Math. Soc. 54(4), 352–360 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  14. Najt, E., Deford, D., Solomon, J.: Complexity and geometry of sampling connected graph partitions. arXiv preprint arXiv:1908.08881 (2019)

  15. Spielman, D.A., Teng, S.H.: Disk packings and planar separators. In: Proceedings of the Twelfth Annual Symposium on Computational Geometry, pp. 349–358 (1996)

    Google Scholar 

  16. Tarjan, R.: Enumeration of the elementary circuits of a directed graph. SIAM J. Comput. 2(3), 211–216 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tiernan, J.C.: An efficient search algorithm to find the elementary circuits of a graph. Commun. ACM 13(12), 722–726 (1970)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We sincerely thank the three anonymous referees for their careful reading of the paper and helpful suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ashwin Maran .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cai, JY., Maran, A. (2022). Counting Cycles on Planar Graphs in Subexponential Time. In: Zhang, Y., Miao, D., Möhring, R. (eds) Computing and Combinatorics. COCOON 2022. Lecture Notes in Computer Science, vol 13595. Springer, Cham. https://doi.org/10.1007/978-3-031-22105-7_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-22105-7_24

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22104-0

  • Online ISBN: 978-3-031-22105-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics