Improved FPTAS for Multi-spin Systems

We design deterministic fully polynomial-time approximation scheme (FPTAS) for computing the partition function for a class of multi-spin systems, extending the known approximable regime by an exponential scale. As a consequence, we have an FPTAS for the Potts models with inverse temperature β up to a critical threshold \(|\beta|=O(\frac{1}{\Delta})\) where Δ is the maximum degree, confirming a conjecture in [10]. We also give an improved FPTAS for a generalization of counting q-colorings, namely the counting list-colorings. As a consequence we have an FPTAS for counting q-colorings in graphs with maximum degree Δ when q ≥ αΔ + 1 for α greater than α * ≈ 2.58071. This is so far the best bound achieved by deterministic approximation algorithms for counting q-colorings. All these improvements are obtained by applying a potential analysis to the correlation decay on computation trees for multi-spin systems.

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