Abstract
We give fully polynomial-time approximation schemes (FPTAS) for the partition function of ferromagnetic 2-spin systems in certain parameter regimes. The threshold we obtain is almost tight up to an integrality gap. Our technique is based on the correlation decay framework. The main technical contribution is a new potential function, with which we establish a new kind of spatial mixing.
- David H. Ackley, Geoffrey E. Hinton, and Terrence J. Sejnowski. 1985. A learning algorithm for Boltzmann machines. Cogn. Sci. 9, 1 (1985), 147--169.Google Scholar
Cross Ref
- Jin-Yi Cai, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Mark Jerrum, Daniel Štefankovič, and Eric Vigoda. 2016. #BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region. J. Comput. Syst. Sci. 82, 5 (2016), 690--711. Google Scholar
Digital Library
- Jin-Yi Cai and Michael Kowalczyk. 2012. Spin systems on k-regular graphs with complex edge functions. Theor. Comput. Sci. 461 (2012), 2--16. Google Scholar
Digital Library
- Martin E. Dyer, Leslie Ann Goldberg, Catherine S. Greenhill, and Mark Jerrum. 2003. The relative complexity of approximate counting problems. Algorithmica 38, 3 (2003), 471--500.Google Scholar
Cross Ref
- Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. 2016. Inapproximability of the partition function for the antiferromagnetic Ising and hard-core models. Comb. Probab. Comput. 25, 4 (2016), 500--559.Google Scholar
Cross Ref
- Hans-Otto Georgii. 2011. Gibbs Measures and Phase Transitions (2nd ed.). De Gruyter Studies in Mathematics, Vol. 9. de Gruyter, Berlin.Google Scholar
- Leslie Ann Goldberg and Mark Jerrum. 2007. The complexity of ferromagnetic Ising with local fields. Comb. Probab. Comput. 16, 1 (2007), 43--61. Google Scholar
Digital Library
- Leslie Ann Goldberg, Mark Jerrum, and Mike Paterson. 2003. The computational complexity of two-state spin systems. Random Struct. Algor. 23, 2 (2003), 133--154.Google Scholar
Cross Ref
- Robert B. Griffiths. 1972. Rigorous results and theorems. In Phase Transitions and Critical Phenomena, Cyril Domb and Melville S. Green (Eds.). Vol. 1. Academic Press, 7--109.Google Scholar
- Heng Guo and Pinyan Lu. 2016. Uniqueness, spatial mixing, and approximation for ferromagnetic 2-spin systems. In Proceedings of the RANDOM (LIPIcs), Vol. 60. 31:1--31:26.Google Scholar
- Mark Jerrum and Alistair Sinclair. 1993. Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22, 5 (1993), 1087--1116. Google Scholar
Digital Library
- Mark Jerrum, Leslie G. Valiant, and Vijay V. Vazirani. 1986. Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43 (1986), 169--188. Google Scholar
Cross Ref
- Frank P. Kelly. 1985. Stochastic models of computer communication systems. J. R. Stat. Soc. Series B Stat. Methodol. 47, 3 (1985), 379--395.Google Scholar
Cross Ref
- Liang Li, Pinyan Lu, and Yitong Yin. 2012. Approximate counting via correlation decay in spin systems. In Proceedings of the SODA. 922--940. Google Scholar
Digital Library
- Liang Li, Pinyan Lu, and Yitong Yin. 2013. Correlation decay up to uniqueness in spin systems. In Proceedings of the SODA. 67--84. Google Scholar
Digital Library
- Jingcheng Liu, Pinyan Lu, and Chihao Zhang. 2014. The complexity of ferromagnetic two-spin systems with external fields. In Proceedings of the RANDOM. 843--856.Google Scholar
- Russell Lyons. 1989. The Ising model and percolation on trees and tree-like graphs. Comm. Math. Phys. 125, 2 (1989), 337--353.Google Scholar
Cross Ref
- Elchanan Mossel and Allan Sly. 2013. Exact thresholds for Ising-Gibbs samplers on general graphs. Ann. Probab. 41, 1 (2013), 294--328.Google Scholar
Cross Ref
- Alistair Sinclair, Piyush Srivastava, Daniel Štefankovič, and Yitong Yin. 2017. Spatial mixing and the connective constant: Optimal bounds. Probab. Theory Related Fields 168, 1 (2017), 153--197.Google Scholar
Cross Ref
- Alistair Sinclair, Piyush Srivastava, and Marc Thurley. 2014. Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs. J. Stat. Phys. 155, 4 (2014), 666--686.Google Scholar
Cross Ref
- Allan Sly. 2010. Computational transition at the uniqueness threshold. In Proceedings of the FOCS. 287--296. Google Scholar
Digital Library
- Allan Sly and Nike Sun. 2014. The computational hardness of counting in two-spin models on d-regular graphs. Ann. Probab. 42, 6 (2014), 2383--2416.Google Scholar
Cross Ref
- Dror Weitz. 2006. Counting independent sets up to the tree threshold. In Proceedings of the STOC. 140--149. Google Scholar
Digital Library
- Jinshan Zhang, Heng Liang, and Fengshan Bai. 2011. Approximating partition functions of the two-state spin system. Inf. Process. Lett. 111, 14 (2011), 702--710. Google Scholar
Digital Library
Index Terms
Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems
Recommendations
#BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region
Counting independent sets on bipartite graphs (#BIS) is considered a canonical counting problem of intermediate approximation complexity. It is conjectured that #BIS neither has an FPRAS nor is as hard as #Sat to approximate. We study #BIS in the ...
Approximation via Correlation Decay When Strong Spatial Mixing Fails
Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of antiferromagnetic 2-spin models. Previous analyses ...
Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results
Recent results establish for the hard-core model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree $\Delta$ undergoes a phase transition that ...






Comments