[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

  • Conference paper
  • First Online:
Integer Programming and Combinatorial Optimization (IPCO 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12707))

  • 917 Accesses

Abstract

Spanning trees are a representative example of linear matroid bases that are efficiently countable. Perfect matchings of Pfaffian bipartite graphs are a countable example of common bases of two matrices. Generalizing these two, Webb (2004) introduced the notion of Pfaffian pairs as a pair of matrices for which counting of their common bases is tractable via the Cauchy–Binet formula.

This paper studies counting on linear matroid problems extending Webb’s work. We first introduce “Pfaffian parities” as an extension of Pfaffian pairs to the linear matroid parity problem, which is a common generalization of the linear matroid intersection problem and the matching problem. We show that a large number of efficiently countable discrete structures are interpretable as special cases of Pfaffian pairs and parities.

We also observe that the fastest randomized algorithms for the linear matroid intersection and parity problems by Harvey (2009) and Cheung–Lau–Leung (2014) can be derandomized for Pfaffian pairs and parities. We further present polynomial-time algorithms to count the number of minimum-weight solutions on weighted Pfaffian pairs and parities.

The full version of this paper is available at https://arxiv.org/abs/1912.00620.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /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
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 35.99
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
GBP 44.99
Price includes VAT (United Kingdom)
  • 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.

    An equivalent definition of Pfaffian orientation is as follows: an orientation of G is Pfaffian if every even-length cycle C such that \(G-V(C)\) has a perfect matching has an odd number of edges directed in either direction.

References

  1. Anari, N., Gharan, S.O., Vinzant, C.: Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids. In: Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018), pp. 35–46 (2018). https://doi.org/10.1109/FOCS.2018.00013

  2. Anari, N., Liu, K., Gharan, S.O., Vinzant, C.: Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. In: Proceedings of the 51st Annual ACM Symposium on Theory of Computing (STOC 2019), pp. 1–12 (2019). https://doi.org/10.1145/3313276.3316385

  3. Cheung, H.Y., Lau, L.C., Leung, K.M.: Algebraic algorithms for linear matroid parity problems. ACM Trans. Algorithms 10(3), 1–26 (2014). https://doi.org/10.1145/2601066

    Article  MathSciNet  MATH  Google Scholar 

  4. Colbourn, C.J., Provan, J.S., Vertigan, D.: The complexity of computing the Tutte polynomial on transversal matroids. Combinatorica 15(1), 1–10 (1995). https://doi.org/10.1007/BF01294456

    Article  MathSciNet  MATH  Google Scholar 

  5. Edmonds, J.: Matroid partition. In: Dantzig, G.B., Veinott, Jr., A.F. (eds.) Mathematics of the Decision Sciences: Part I, Lectures in Applied Mathematics, vol. 11, pp. 335–345. AMS, Providence, RI (1968). https://doi.org/10.1007/978-3-540-68279-0_7

  6. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York, NY (1970). https://doi.org/10.1007/3-540-36478-1_2

  7. Frank, A.: Connections in Combinatorial Optimization. Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, New York, NY (2011)

    Google Scholar 

  8. Gabow, H.N., Stallmann, M.: An augmenting path algorithm for linear matroid parity. Combinatorica 6(2), 123–150 (1986). https://doi.org/10.1007/BF02579169

    Article  MathSciNet  MATH  Google Scholar 

  9. Gabow, H.N., Xu, Y.: Efficient theoretic and practical algorithms for linear matroid intersection problems. J. Comput. Syst. Sci. 53(1), 129–147 (1996). https://doi.org/10.1006/jcss.1996.0054

    Article  MathSciNet  MATH  Google Scholar 

  10. Gallai, T.: Maximum-Minimum Sätze und verallgemeinerte Faktoren von Graphen. Acta Mathematica Academiae Scientiarum Hungaricae 12, 131–173 (1964). https://doi.org/10.1007/BF02066678

  11. Gessel, I., Viennot, G.: Binomial determinants, paths, and hook length formulae. Adv. Math. 58(3), 300–321 (1985). https://doi.org/10.1016/0001-8708(85)90121-5

    Article  MathSciNet  MATH  Google Scholar 

  12. Goodall, A., De Mier, A.: Spanning trees of 3-uniform hypergraphs. Adv. Appl. Math. 47(4), 840–868 (2011). https://doi.org/10.1016/j.aam.2011.04.006

    Article  MathSciNet  MATH  Google Scholar 

  13. Harvey, N.J.A.: Algebraic algorithms for matching and matroid problems. SIAM J. Comput. 39(2), 679–702 (2009). https://doi.org/10.1137/070684008

    Article  MathSciNet  MATH  Google Scholar 

  14. Ishikawa, M., Wakayama, M.: Minor summation formula of Pfaffians. Linear Multilinear Algebra 39(3), 285–305 (1995). https://doi.org/10.1080/03081089508818403

    Article  MathSciNet  MATH  Google Scholar 

  15. Iwata, S., Kobayashi, Y.: A weighted linear matroid parity algorithm. SIAM J. Comput. (to appear). https://doi.org/10.1137/17M1141709

  16. Kasteleyn, P.W.: The statistics of dimers on a lattice: I. the number of dimer arrangements on a quadratic lattice. Physica, 27(12), 1209–1225 (1961). https://doi.org/10.1016/0031-8914(61)90063-5

  17. Kasteleyn, P.W.: Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 43–110. Academic Press, New York, NY (1967)

    MATH  Google Scholar 

  18. Kirchhoff, G.: Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Annalen der Physik 148(12), 497–508 (1847). https://doi.org/10.1002/andp.18471481202

    Article  Google Scholar 

  19. Lawler, E.L.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York, NY (1976)

    MATH  Google Scholar 

  20. Lindström, B.: On the vector representations of induced matroids. Bull. London Math. Soc. 5(1), 85–90 (1973). https://doi.org/10.1112/blms/5.1.85

    Article  MathSciNet  MATH  Google Scholar 

  21. Little, C.H.C.: An extension of Kasteleyn’s method of enumerating the 1-factors of planar graphs. In: Holton, D.A. (ed.) Combinatorial Mathematics. LNM, vol. 403, pp. 63–72. Springer, Heidelberg (1974). https://doi.org/10.1007/BFb0057377

    Chapter  Google Scholar 

  22. Lovász, L.: Matroid matching and some applications. J. Comb. Theor. Ser. B 28(2), 208–236 (1980). https://doi.org/10.1016/0095-8956(80)90066-0

    Article  MathSciNet  MATH  Google Scholar 

  23. Mader, W.: Über die Maximalzahl kreuzungsfreier \(H\)-Wege. Archiv der Mathematik 31(1), 387–402 (1978). https://doi.org/10.1007/BF01226465

    Article  MathSciNet  MATH  Google Scholar 

  24. Maurer, S.B.: Matrix generalizations of some theorems on trees, cycles and cocycles in graphs. SIAM J. Appl. Math. 30(1), 143–148 (1976). https://doi.org/10.1137/0130017

    Article  MathSciNet  MATH  Google Scholar 

  25. Murota, K.: Computing the degree of determinants via combinatorial relaxation. SIAM J. Comput. 24(4), 765–796 (1995)

    Article  MathSciNet  Google Scholar 

  26. Orlin, J.B.: A fast, simpler algorithm for the matroid parity problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 240–258. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68891-4_17

    Chapter  Google Scholar 

  27. Robertson, N., Seymour, P.D., Thomas, R.: Permanents, Pfaffian orientations, and even directed circuits. Ann. Math. 150(3), 929–975 (1999). https://doi.org/10.2307/121059

    Article  MathSciNet  MATH  Google Scholar 

  28. Schrijver, A.: Combinatorial Optimization, Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)

    MATH  Google Scholar 

  29. Snook, M.: Counting bases of representable matroids. Electron. J. Comb. 19(4), P41 (2012)

    Google Scholar 

  30. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics-an exact result. Philos. Mag. 6(68), 1061–1063 (1961). https://doi.org/10.1080/14786436108243366

    Article  MathSciNet  MATH  Google Scholar 

  31. Tutte, W.T.: The dissection of equilateral triangles into equilateral triangles. Math. Proc. Camb. Philos. Soc. 44(4), 463–482 (1948). https://doi.org/10.1017/S030500410002449X

    Article  MathSciNet  MATH  Google Scholar 

  32. Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8(2), 189–201 (1979). https://doi.org/10.1016/0304-3975(79)90044-6

    Article  MathSciNet  MATH  Google Scholar 

  33. Vazirani, V.V.: NC algorithms for computing the number of perfect matchings in \(K_{3,3}\)-free graphs and related problems. Inf. Comput. 80(2), 152–164 (1989). https://doi.org/10.1016/0890-5401(89)90017-5

    Article  MATH  Google Scholar 

  34. Webb, K.P.: Counting Bases. Ph.D. thesis, University of Waterloo, Waterloo, ON (2004)

    Google Scholar 

  35. Yamaguchi, Y.: Shortest disjoint \(\cal{S}\)-paths via weighted linear matroid parity. In: Hong, S.H. (ed.) Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC ’16). Leibniz International Proceedings in Informatics, vol. 64, pp. 63:1–63:13. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2016). https://doi.org/10.4230/LIPIcs.ISAAC.2016.63

Download references

Acknowledgments

The authors thank Satoru Iwata for his helpful comments, and Yusuke Kobayashi, Yutaro Yamaguchi, and Koyo Hayashi for discussions. This work was supported by JST ACT-I Grant Number JPMJPR18U9, Japan, and Grant-in-Aid for JSPS Research Fellow Grant Number JP18J22141, Japan.

Author information

Authors and Affiliations

  1. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo, 113-8656, Japan

    Kazuki Matoya & Taihei Oki

Authors
  1. Kazuki Matoya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Taihei Oki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taihei Oki .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Mohit Singh

  2. Cornell University, Ithaca, NY, USA

    David P. Williamson

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Matoya, K., Oki, T. (2021). Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems. In: Singh, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2021. Lecture Notes in Computer Science(), vol 12707. Springer, Cham. https://doi.org/10.1007/978-3-030-73879-2_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-73879-2_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-73878-5

  • Online ISBN: 978-3-030-73879-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation