The Complexity of Approximating the Matching Polynomial in the Complex Plane

Authors Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Daniel Štefankovič



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Ivona Bezáková
  • Department of Computer Science, Rochester Institute of Technology, Rochester, NY, USA
Andreas Galanis
  • Department of Computer Science, University of Oxford, UK
Leslie Ann Goldberg
  • Department of Computer Science, University of Oxford, UK
Daniel Štefankovič
  • Department of Computer Science, University of Rochester, Rochester, NY, USA

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Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, and Daniel Štefankovič. The Complexity of Approximating the Matching Polynomial in the Complex Plane. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.22

Abstract

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter gamma, where gamma takes arbitrary values in the complex plane. When gamma is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of gamma, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Delta as long as gamma is not a negative real number less than or equal to -1/(4(Delta-1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Delta >= 3 and all real gamma less than -1/(4(Delta-1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Delta with edge parameter gamma is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real gamma it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of gamma values on the negative real axis. Nevertheless, we show that the result does extend for any complex value gamma that does not lie on the negative real axis. Our analysis accounts for complex values of gamma using geodesic distances in the complex plane in the metric defined by an appropriate density function.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • matchings
  • partition function
  • correlation decay
  • connective constant

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References

  1. A. Barvinok. Computing the Permanent of (Some) Complex Matrices. Foundations of Computational Mathematics, 16(2):329-342, 2016. Google Scholar
  2. A. Barvinok. Combinatorics and Complexity of Partition Functions. Algorithms and Combinatorics. Springer International Publishing, 2017. Google Scholar
  3. M. Bayati, D. Gamarnik, D. A. Katz, C. Nair, and P. Tetali. Simple Deterministic Approximation Algorithms for Counting Matchings. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 122-127, 2007. Google Scholar
  4. I. Bezáková, A. Galanis, L. A. Goldberg, and D. Štefankovič. Inapproximability of the Independent Set Polynomial in the Complex Plane. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1234-1240, 2018. Full version available from URL: https://arxiv.org/abs/1711.00282.
  5. J.-Y. Cai, S. Huang, and P. Lu. From Holant to #CSP and Back: Dichotomy for Holant^c Problems. Algorithmica, 64(3):511-533, 2012. Google Scholar
  6. C. D. Godsil. Matchings and Walks in Graphs. Journal of Graph Theory, 5(3):285-297, 1981. Google Scholar
  7. Nicholas J. A. Harvey, Piyush Srivastava, and Jan Vondrák. Computing the Independence Polynomial: from the Tree Threshold down to the Roots. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, pages 1557-1576, 2018. Google Scholar
  8. O. J. Heilmann and E. H. Lieb. Theory of Monomer-Dimer Systems. Communications in Mathematical Physics, 25(3):190-232, 1972. Google Scholar
  9. M. Jerrum and A. Sinclair. Approximating the Permanent. SIAM J. Comput., 18(6):1149-1178, 1989. Google Scholar
  10. D. Kraus and O. Roth. Conformal Metrics. CoRR, abs/0805.2235, 2008. URL: http://arxiv.org/abs/0805.2235.
  11. V. Patel and G. Regts. Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials. SIAM Journal on Computing, 46(6):1893-1919, 2017. Google Scholar
  12. H. Peters and G. Regts. On a Conjecture of Sokal Concerning Roots of the Independence Polynomial. The Michigan Mathematical Journal, 2019. To appear. URL: https://projecteuclid.org/euclid.mmj/1541667626#info.
  13. A. Sinclair, P. Srivastava, D. Štefankovič, and Y. Yin. Spatial Mixing and the Connective Constant: Optimal Bounds. Probability Theory and Related Fields, 168(1):153-197, 2017. Google Scholar
  14. A. Sinclair, P. Srivastava, and Y. Yin. Spatial Mixing and Approximation Algorithms for Graphs with Bounded Connective Constant. In Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS '13, pages 300-309, 2013. Google Scholar
  15. B.-B. Wei, S.-W. Chen, H.-C. Po, and R.-B. Liu. Phase Transitions in the Complex Plane of Physical Parameters. Nature Scientific Reports, 4:5202; DOI: 10.1038/srep05202, 2014. Google Scholar
  16. D. Weitz. Counting Independent Sets Up to the Tree Threshold. In Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, STOC '06, pages 140-149, New York, NY, USA, 2006. ACM. Google Scholar