Counting Hypergraph Matchings up to Uniqueness Threshold

Authors Renjie Song, Yitong Yin, Jinman Zhao

Thumbnail PDF


  • Filesize: 1.86 MB
  • 29 pages

Document Identifiers

Author Details

Renjie Song
Yitong Yin
Jinman Zhao

Cite AsGet BibTex

Renjie Song, Yitong Yin, and Jinman Zhao. Counting Hypergraph Matchings up to Uniqueness Threshold. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 46:1-46:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter lambda, each matching M is assigned a weight lambda^{|M|}. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). For this model, the critical activity lambda_c= (d^d)/(k (d-1)^{d+1}) is the threshold for the uniqueness of Gibbs measures on the infinite (d+1)-uniform (k+1)-regular hypertree. Consider hypergraphs of maximum degree at most k+1 and maximum size of hyperedges at most d+1. We show that when lambda < lambda_c, there is an FPTAS for computing the partition function; and when lambda = lambda_c, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When lambda > 2lambda_c, there is no PRAS for the partition function or the log-partition function unless NP=RP. Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets.
  • approximate counting; phase transition; spatial mixing


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Mohsen Bayati, David Gamarnik, Dimitriy Katz, Chandra Nair, and Prasad Tetali. Simple deterministic approximation algorithms for counting matchings. In Proceedings of the 39th ACM Symposium on Theory of Computing (STOC), pages 122-127, 2007. Google Scholar
  2. Ivona Bezakova, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Daniel Stefankovic. Counting independent sets in hypergraphs when strong spatial mixing fails. arXiv preprint arXiv:1510.09193, 2015. Google Scholar
  3. Magnus Bordewich, Martin Dyer, and Marek Karpinski. Path coupling using stopping times and counting independent sets and colorings in hypergraphs. Random Structures &Algorithms, 32(3):375-399, 2008. Google Scholar
  4. Amir Dembo, Andrea Montanari, et al. Ising models on locally tree-like graphs. The Annals of Applied Probability, 20(2):565-592, 2010. Google Scholar
  5. Andrzej Dudek, Marek Karpinski, Andrzej Ruciński, and Edyta Szymańska. Approximate counting of matchings in (3, 3)-hypergraphs. In SWAT, pages 380-391, 2014. Google Scholar
  6. Martin Dyer, Leslie A Goldberg, and Mark Jerrum. Counting and sampling H-colourings. In RANDOM, pages 51-67, 2002. Google Scholar
  7. Andreas Galanis, Qi Ge, Daniel Štefankovič, Eric Vigoda, and Linji Yang. Improved inapproximability results for counting independent sets in the hard-core model. Random Structures &Algorithms, 45(1):78-110, 2014. Google Scholar
  8. Andreas Galanis and Leslie Ann Goldberg. The complexity of approximately counting in 2-spin systems on k-uniform bounded-degree hypergraphs. arXiv preprint arXiv:1505.06146, 2015. Google Scholar
  9. Andreas Galanis, Daniel Stefankovic, and Eric Vigoda. Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. arXiv preprint arXiv:1203.2226, 2012. Google Scholar
  10. Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region. In Proceedings of the 46th ACM Symposium on Theory of Computing (STOC), pages 823-831, 2014. Google Scholar
  11. David Gamarnik and Dmitriy Katz. Correlation decay and deterministic FPTAS for counting colorings of a graph. Journal of Discrete Algorithms, 12:29-47, 2012. Google Scholar
  12. David Gamarnik, Dmitriy Katz, and Sidhant Misra. Strong spatial mixing of list coloring of graphs. Random Structures &Algorithms, 2013. Google Scholar
  13. Ole J Heilmann. Existence of phase transitions in certain lattice gases with repulsive potential. Lettere Al Nuovo Cimento (1971-1985), 3(3):95-98, 1972. Google Scholar
  14. Ole J Heilmann and Elliott H Lieb. Theory of monomer-dimer systems. Communications in Mathematical Physics, 25(3):190-232, 1972. Google Scholar
  15. Svante Janson, Tomasz Luczak, and Andrzej Rucinski. Random graphs, volume 45. John Wiley &Sons, 2011. Google Scholar
  16. Mark Jerrum and Alistair Sinclair. Approximating the permanent. SIAM Journal on Computing, 18(6):1149-1178, 1989. Google Scholar
  17. Marek Karpinski, Andrzej Rucinski, and Edyta Szymanska. Approximate counting of matchings in sparse uniform hypergraphs. In Proceedings of the Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pages 72-79, 2013. Google Scholar
  18. Frank P Kelly. Stochastic models of computer communication systems. Journal of the Royal Statistical Society. Series B (Methodological), 47(3):379-395, 1985. Google Scholar
  19. Liang Li, Pinyan Lu, and Yitong Yin. Approximate counting via correlation decay in spin systems. In Proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 922-940, 2012. Google Scholar
  20. Liang Li, Pinyan Lu, and Yitong Yin. Correlation decay up to uniqueness in spin systems. In Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 67-84, 2013. Google Scholar
  21. Chengyu Lin, Jingcheng Liu, and Pinyan Lu. A simple FPTAS for counting edge covers. In Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 341-348, 2014. Google Scholar
  22. Jingcheng Liu and Pinyan Lu. FPTAS for counting monotone CNF. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1531-1548, 2015. Google Scholar
  23. Pinyan Lu, Menghui Wang, and Chihao Zhang. FPTAS for weighted fibonacci gates and its applications. In Proceedings of the 41st International Colloquium on Automata, Languages and Programming (ICALP), pages 787-799, 2014. Google Scholar
  24. Pinyan Lu, Kuan Yang, and Chihao Zhang. Fptas for hardcore and ising models on hypergraphs. arXiv preprint arXiv:1509.05494, 2015. Google Scholar
  25. Pinyan Lu and Yitong Yin. Improved FPTAS for multi-spin systems. In RANDOM, pages 639-654, 2013. Google Scholar
  26. Elchanan Mossel, Dror Weitz, and Nicholas Wormald. On the hardness of sampling independent sets beyond the tree threshold. Probability Theory and Related Fields, 143(3-4):401-439, 2009. Google Scholar
  27. Ricardo Restrepo, Jinwoo Shin, Prasad Tetali, Eric Vigoda, and Linji Yang. Improved mixing condition on the grid for counting and sampling independent sets. Probability Theory and Related Fields, 156(1-2):75-99, 2013. Google Scholar
  28. Alistair Sinclair, Piyush Srivastava, and Marc Thurley. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. Journal of Statistical Physics, 155(4):666-686, 2014. Google Scholar
  29. Alistair Sinclair, Piyush Srivastava, and Yitong Yin. Spatial mixing and approximation algorithms for graphs with bounded connective constant. In Proceedings of the 54th IEEE Symposium on Foundations of Computer Science (FOCS), pages 300-309, 2013. Google Scholar
  30. Allan Sly. Computational transition at the uniqueness threshold. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science (FOCS), pages 287-296, 2010. Google Scholar
  31. Allan Sly, Nike Sun, et al. Counting in two-spin models on d-regular graphs. The Annals of Probability, 42(6):2383-2416, 2014. Google Scholar
  32. Frank Spitzer. Markov random fields on an infinite tree. The Annals of Probability, pages 387-398, 1975. Google Scholar
  33. Dror Weitz. Combinatorial criteria for uniqueness of Gibbs measures. Random Structures &Algorithms, 27(4):445-475, 2005. Google Scholar
  34. Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the 38th ACM Symposium on Theory of Computing (STOC), pages 140-149, 2006. Google Scholar