Approximation via Correlation Decay when Strong Spatial Mixing Fails
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 (SSM). This means that one uses worst-case analysis of the recursive procedure that creates the sub-instances. 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 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 SSM 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 SSM 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>=Delta. This bound is not substantially stronger than existing randomised results in the literature. Nevertheless, it gives the first deterministic approximation scheme in this regime. Moreover, unlike existing results, it leads to an FPTAS for counting dominating sets in regular graphs with sufficiently large degree. We further demonstrate that approximately counting independent sets in hypergraphs is NP-hard even within the uniqueness regime.