Approximate Counting for Spin Systems in Sub-Quadratic Time
(2306.14867)Abstract
We present two approximate counting algorithms with $\widetilde{O}(n{2-c}/\varepsilon2)$ running time for some constant $c > 0$ and accuracy $\varepsilon$: (1) for the hard-core model when strong spatial mixing (SSM) is sufficiently fast; (2) for spin systems with SSM on planar graphs with quadratic growth, such as $\mathbb{Z}2$. The latter algorithm also extends to (not necessarily planar) graphs with polynomial growth, such as $\mathbb{Z}d$, albeit with a running time of the form $\widetilde{O}(n2\varepsilon{-2}/2{c(\log n){1/d}})$ for some constant $c > 0$ and $d$ being the exponent of the polynomial growth. Our technique utilizes Weitz's self-avoiding walk tree (STOC, 2006) and the recent marginal sampler of Anand and Jerrum (SIAM J. Comput., 2022).
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