Andreas Galanis

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Recent inapproximability results of Sly ( 2010 ), together with an approximation algorithm presented by Weitz ( 2006 ), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λ c (

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

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Improved inapproximability results for counting independent sets in the hard-core model

) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented an FPTAS for the partition function when λ c (

Inapproximability of the Independent Set Polynomial Below the Shearer Threshold.

\mathbb{T}_{\Delta}

Swendsen‐Wang algorithm on the mean‐field Potts model

) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists e Δ > 0 such that (unless RP = NP) there is no FPRAS for approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λ c (

The complexity of approximately counting in 2-spin systems on k-uniform bounded-degree hypergraphs

\mathbb{T}_{\Delta}

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

) c (

Inapproximability of the independent set polynomial in the complex plane

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Approximately Counting

) + e Δ . We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava and Thurley ( 2014 ) extended Weitzs approach to the antiferromagnetic Ising model, yielding an FPTAS for the partition function for all graphs of constant maximum degree Δ when the parameters of the model lie in the uniqueness region of the infinite Δ-regular tree. We prove the complementary result for the antiferromagnetic Ising model without external field, namely, that unless RP = NP, for all Δ ⩾ 3, there is no FPRAS for approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree

H

\mathbb{T}_{\Delta}