FPTAS for Hardcore and Ising Models on Hypergraphs
Hardcore and Ising models are two most important families of two state spin systems in statistic physics. Partition function of spin systems is the center concept in statistic physics which connects microscopic particles and their interactions with their macroscopic and statistical properties of materials such as energy, entropy, ferromagnetism, etc. If each local interaction of the system involves only two particles, the system can be described by a graph. In this case, fully polynomial-time approximation scheme (FPTAS) for computing the partition function of both hardcore and anti-ferromagnetic Ising model was designed up to the uniqueness condition of the system. These result are the best possible since approximately computing the partition function beyond this threshold is NP-hard. In this paper, we generalize these results to general physics systems, where each local interaction may involves multiple particles. Such systems are described by hypergraphs. For hardcore model, we also provide FPTAS up to the uniqueness condition, and for anti-ferromagnetic Ising model, we obtain FPTAS under a slightly stronger condition.
hard-core model
ising model
hypergraph
spatial mixing
correlation decay
51:1-51:14
Regular Paper
Pinyan
Lu
Pinyan Lu
Kuan
Yang
Kuan Yang
Chihao
Zhang
Chihao Zhang
10.4230/LIPIcs.STACS.2016.51
Antar Bandyopadhyay and David Gamarnik. Counting without sampling: Asymptotics of the log-partition function for certain statistical physics models. Random Structures &Algorithms, 33(4):452-479, 2008.
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. URL: http://dx.doi.org/10.1002/rsa.20204.
http://dx.doi.org/10.1002/rsa.20204
Martin Dyer, Mark Jerrum, and Eric Vigoda. Rapidly mixing Markov chains for dismantleable constraint graphs. In Randomization and Approximation Techniques in Computer Science, pages 68-77. Springer, 2002.
Martin E. Dyer, Alan M. Frieze, and Mark Jerrum. On counting independent sets in sparse graphs. SIAM Jounal on Computing, 31(5):1527-1541, 2002. URL: http://epubs.siam.org/sam-bin/dbq/article/38384.
http://epubs.siam.org/sam-bin/dbq/article/38384
Martin E. Dyer and Catherine S. Greenhill. On Markov chains for independent sets. Journal of Algorithms, 35(1):17-49, 2000.
A. Galanis, D. Stefankovic, and E. Vigoda. Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Arxiv preprint arXiv:1203.2226, 2012.
David Gamarnik and Dmitriy Katz. Correlation decay and deterministic FPTAS for counting colorings of a graph. Journal of Discrete Algorithms, 12:29-47, 2012.
Leslie Ann Goldberg and Mark Jerrum. A polynomial-time algorithm for estimating the partition function of the ferromagnetic Ising model on a regular matroid. SIAM Journal on Computing, 42(3):1132-1157, 2013.
Mark Jerrum. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures &Algorithms, 7(2):157-166, 1995.
Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on Computing, 22(5):1087-1116, 1993.
Mark Jerrum and Alistair Sinclair. The Markov chain monte carlo method: an approach to approximate counting and integration. Approximation algorithms for NP-hard problems, pages 482-520, 1996.
Mark Jerrum, Alistair Sinclair, and Eric Vigoda. A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. Journal of the ACM, 51:671-697, July 2004. URL: http://dx.doi.org/10.1145/1008731.1008738.
http://dx.doi.org/10.1145/1008731.1008738
Liang Li, Pinyan Lu, and Yitong Yin. Approximate counting via correlation decay in spin systems. In Proceedings of the 23th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'12), pages 922-940. SIAM, 2012.
Liang Li, Pinyan Lu, and Yitong Yin. Correlation decay up to uniqueness in spin systems. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'13), pages 67-84. SIAM, 2013.
Chengyu Lin, Jingcheng Liu, and Pinyan Lu. A simple FPTAS for counting edge covers. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'14), pages 341-348, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.25.
http://dx.doi.org/10.1137/1.9781611973402.25
Jingcheng Liu and Pinyan Lu. FPTAS for #BIS with degree bounds on one side. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC'15), 2015.
Jingcheng Liu and Pinyan Lu. FPTAS for counting monotone CNF. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'15), pages 1531-1548, 2015.
Jingcheng Liu, Pinyan Lu, and Chihao Zhang. FPTAS for counting weighted edge covers. In In Proceedings of the 22nd European Symposium on Algorithms (ESA'14), pages 654-665, 2014.
Pinyan Lu, Menghui Wang, and Chihao Zhang. FPTAS for weighted Fibonacci gates and its applications. Proceedings of the 41st International Colloquium on Automata, Languages and Programming (ICALP'14), pages 787-799, 2014.
Pinyan Lu and Yitong Yin. Improved FPTAS for multi-spin systems. In Proceedings of APPROX-RANDOM, pages 639-654. Springer, 2013.
Michael Luby and Eric Vigoda. Approximately counting up to four. In Proceedings of the 29th Annual ACM Symposium on Theory of computing (STOC'97), pages 682-687. ACM, 1997.
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.
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.
Allan Sly and Nike Sun. The computational hardness of counting in two-spin models on d-regular graphs. In Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS'12), pages 361-369. IEEE, 2012.
Eric Vigoda. Improved bounds for sampling colorings. Journal of Mathematical Physics, 41(3):1555-1569, 2000.
Dror Weitz. Counting independent sets up to the tree threshold. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC'06), pages 140-149. ACM, 2006.
Yitong Yin and Jinman Zhao. Counting hypergraph matchings up to uniqueness threshold. arXiv preprint arXiv:1503.05812, 2015.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode