Danielštefankovič ‡

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

[1]  Mark Jerrum,et al.  The Swendsen-Wang process does not always mix rapidly , 1997, STOC '97.

[2]  Christian Borgs,et al.  Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point , 2010, ArXiv.

[3]  Olle Häggström,et al.  The random-cluster model on a homogeneous tree , 1996 .

[4]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[5]  Liang Li,et al.  Correlation Decay up to Uniqueness in Spin Systems , 2013, SODA.

[6]  S. Zachary,et al.  Loss networks , 2009, 0903.0640.

[7]  Eric Vigoda,et al.  Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models , 2012, Combinatorics, Probability and Computing.

[8]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[9]  Svante Janson,et al.  Random Regular Graphs: Asymptotic Distributions and Contiguity , 1995, Combinatorics, Probability and Computing.

[10]  Martin E. Dyer,et al.  The expressibility of functions on the boolean domain, with applications to counting CSPs , 2011, JACM.

[11]  Leslie Ann Goldberg,et al.  Approximating the partition function of the ferromagnetic Potts model , 2010, JACM.

[12]  Dror Weitz,et al.  Counting independent sets up to the tree threshold , 2006, STOC '06.

[13]  A. Sinclair,et al.  Glauber Dynamics on Trees: Boundary Conditions and Mixing Time , 2003, math/0307336.

[14]  A. Frieze,et al.  Mixing properties of the Swendsen-Wang process on classes of graphs , 1999, Random Struct. Algorithms.

[15]  Olle Hggstrm The random-cluster model on a homogeneous tree , 1996 .

[16]  Allan Sly,et al.  Communications in Mathematical Physics The Replica Symmetric Solution for Potts Models on d-Regular Graphs , 2022 .

[17]  Elchanan Mossel,et al.  On the hardness of sampling independent sets beyond the tree threshold , 2007, math/0701471.

[18]  Eric Vigoda,et al.  Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region , 2013, STOC.

[19]  Eric Vigoda,et al.  Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model , 2011, APPROX-RANDOM.

[20]  Elchanan Mossel,et al.  Survey: Information Flow on Trees , 2004 .

[21]  Martin E. Dyer,et al.  The Relative Complexity of Approximate Counting Problems , 2000, Algorithmica.

[22]  Fabio Martinelli,et al.  Fast mixing for independent sets, colorings, and other models on trees , 2004, SODA '04.

[23]  Allan Sly,et al.  The Computational Hardness of Counting in Two-Spin Models on d-Regular Graphs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[24]  F. Martinelli Rapid mixing of Swendsen-Wang dynamics in two dimensions , 2012 .

[25]  Alan M. Frieze,et al.  Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[26]  Martin E. Dyer,et al.  The complexity of approximating conservative counting CSPs , 2013, STACS.

[27]  Leslie Ann Goldberg,et al.  The computational complexity of two‐state spin systems , 2003, Random Struct. Algorithms.

[28]  Nicholas C. Wormald,et al.  Almost All Regular Graphs Are Hamiltonian , 1994, Random Struct. Algorithms.

[29]  Ravi Montenegro,et al.  Mathematical Aspects of Mixing Times in Markov Chains , 2006, Found. Trends Theor. Comput. Sci..

[30]  Geoffrey Grimmett The Random-Cluster Model , 2002, math/0205237.

[31]  V. Bapst,et al.  The Condensation Phase Transition in Random Graph Coloring , 2016 .

[32]  Eric Vigoda,et al.  #BIS-hardness for 2-spin systems on bipartite bounded degree graphs in the tree non-uniqueness region , 2013, J. Comput. Syst. Sci..

[33]  Martin E. Dyer,et al.  On the relative complexity of approximate counting problems , 2000, APPROX.

[34]  A. Dembo,et al.  Ising models on locally tree-like graphs , 2008, 0804.4726.

[35]  G. Brightwell,et al.  Random colorings of a cayley tree , 2002 .

[36]  Allan Sly,et al.  The number of solutions for random regular NAE-SAT , 2016, Probability Theory and Related Fields.

[37]  Andrea Montanari,et al.  Factor models on locally tree-like graphs , 2011, ArXiv.

[38]  Allan Sly,et al.  Computational Transition at the Uniqueness Threshold , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[39]  F. Peruggi,et al.  Phase diagrams of the q-state potts model on Bethe lattices , 1987 .

[40]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for the Ising Model , 1990, SIAM J. Comput..

[41]  Eric Vigoda,et al.  Improved inapproximability results for counting independent sets in the hard-core model , 2014, Random Struct. Algorithms.

[42]  Mario Ullrich Rapid mixing of Swendsen–Wang dynamics in two dimensions , 2012, 1212.4908.

[43]  N. Wormald,et al.  On the chromatic number of random d-regular graphs , 2008, 0812.2937.

[44]  Richard S. Ellis,et al.  Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model , 2005 .

[45]  Elchanan Mossel,et al.  Exact thresholds for Ising–Gibbs samplers on general graphs , 2009, The Annals of Probability.

[46]  Andrea Montanari,et al.  Reconstruction for Models on Random Graphs , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[47]  Pinyan Lu,et al.  The Complexity of Ferromagnetic Two-spin Systems with External Fields , 2014, APPROX-RANDOM.

[48]  Allan Sly,et al.  Glauber Dynamics of colorings on trees , 2014 .

[49]  Eric Vigoda,et al.  Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results , 2014, APPROX-RANDOM.

[50]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[51]  Catherine S. Greenhill The complexity of counting colourings and independent sets in sparse graphs and hypergraphs , 2000, computational complexity.