NDLI: Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region

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Inapproximability for antiferromagnetic spin systems in the tree non-uniqueness region

Content Provider	ACM Digital Library
Author	Vigoda, Eric Štefankovič, Daniel Galanis, Andreas
Abstract	A remarkable connection has been established for antiferromagnetic 2-spin systems, including the Ising and hard-core models, showing that the computational complexity of approximating the partition function for graphs with maximum degree Δ undergoes a phase transition that coincides with the statistical physics uniqueness/non-uniqueness phase transition on the infinite Δ-regular tree. Despite this clear picture for 2-spin systems, there is little known for multi-spin systems. We present the first analog of the above inapproximability results for multi-spin systems. The main difficulty in previous inapproximability results was analyzing the behavior of the model on random Δ-regular bipartite graphs, which served as the gadget in the reduction. To this end one needs to understand the moments of the partition function. Our key contribution is connecting: (i) induced matrix norms, (ii) maxima of the expectation of the partition function, and (iii) attractive fixed points of the associated tree recursions (belief propagation). The view through matrix norms allows a simple and generic analysis of the second moment for any spin system on random Δ-regular bipartite graphs. This yields concentration results for any spin system in which one can analyze the maxima of the first moment. The connection to fixed points of the tree recursions enables an analysis of the maxima of the first moment for specific models of interest. For k-colorings we prove that for even k, in the tree non-uniqueness region (specifically for semi-translation invariant measures which corresponds to k < Δ) it is NP-hard, unless NP=RP, to approximate the number of colorings for triangle-free Δ-regular graphs. Our proof extends to the antiferromagnetic Potts model, and, in fact, to every antiferromagnetic model under a mild condition.
Starting Page	823
Ending Page	831
Page Count	9
File Format	PDF MP4
ISBN	9781450327107
DOI	10.1145/2591796.2591878
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2014-05-31
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Colorings Potts model Phase transitions Approximate counting
Content Type	Audio Text
Resource Type	Article

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