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# Approximation via Correlation Decay When Strong Spatial Mixing Fails

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LIPIcs.ICALP.2016.45.pdf
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## Cite As

Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Daniel Stefankovic. Approximation via Correlation Decay When Strong Spatial Mixing Fails. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.45

## Abstract

Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of anti-ferromagnetic two-spin models. Previous analyses of correlation-decay algorithms implicitly depended on the occurrence of strong spatial mixing. This, roughly, means that one uses worst-case analysis of the recursive procedure that creates the sub-instances. In this paper, we develop a new analysis method that is more refined than the worst-case analysis. We take the shape of instances in the computation tree into consideration and we amortise against certain "bad" instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain an FPTAS even when strong spatial mixing fails. We apply our technique to the problem of approximately counting independent sets in hypergraphs with degree upper-bound Delta and with a lower bound k on the arity of hyperedges. Liu and Lin gave an FPTAS for k >= 2 and Delta <= 5 (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalised to Delta = 6). Our technique gives a tight result for Delta = 6, showing that there is an FPTAS for k >= 3 and Delta <= 6. The best previously-known approximation scheme for Delta = 6 is the Markov-chain simulation based FPRAS of Bordewich, Dyer and Karpinski, which only works for k >= 8. Our technique also applies for larger values of k, giving an FPTAS for k >= 1.66 Delta. This bound is not as strong as existing randomised results, for technical reasons that are discussed in the paper. Nevertheless, it gives the first deterministic approximation schemes in this regime. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime.
##### Keywords
• approximate counting
• independent sets in hypergraphs
• correlation decay

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