Abstract
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.
Highlights
1.1 Spin SystemsWe study the ferromagnetic Potts model and present tools which are useful for any ferromagnetic spin system on random regular graphs
As a further consequence of our results, we prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing (i. e., exponentially slow convergence to its stationary distribution) on random ∆-regular graphs at the critical temperature for sufficiently large q. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems
Goldberg and Jerrum [20] showed that approximating the partition function of the ferromagnetic Potts model is #BIS-hard, it appears likely that the ferromagnetic Potts model is inapproximable for general graphs
Summary
We study the ferromagnetic Potts model and present tools which are useful for any ferromagnetic spin system on random regular graphs. A specification of a q-state spin model is defined by a symmetric q × q interaction matrix B with non-negative entries. The models are called ferromagnetic if B > 1 since neighboring spins prefer to align and antiferromagnetic if B < 1. The hard-core model is an example of a 2-spin antiferromagnetic system, its interaction matrix is defined so that Ω is the set of independent sets of G and configuration σ ∈ Ω has weight w(σ) = λ|σ| for activity λ > 0. In contrast to the above notion of a ferromagnetic system, in [17] a model is called antiferromagnetic if all of the eigenvalues of B are negative except for the largest (which, as noted above, is positive)
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