اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدApproximate counting and sampling via local central limit theorems
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We give an FPTAS for computing the number of matchings of size $k$ in a graph $G$ of maximum degree $Delta$ on $n$ vertices, for all $k le (1-delta)m^*(G)$, where $delta>0$ is fixed and $m^*(G)$ is the matching number of $G$, and an FPTAS for the number of independent sets of size $k le (1-delta) alpha_c(Delta) n$, where $alpha_c(Delta)$ is the NP-hardness threshold for this problem. We also provide quasi-linear time randomized algorithms to approximately sample from the uniform distribution on matchings of size $k leq (1-delta)m^*(G)$ and independent sets of size $k leq (1-delta)alpha_c(Delta)n$. Our results are based on a new framework for exploiting local central limit theorems as an algorithmic tool. We use a combination of Fourier inversion, probabilistic estimates, and the deterministic approximation of partition functions at complex activities to extract approximations of the coefficients of the partition function. For our results for independent sets, we prove a new local central limit theorem for the hard-core model that applies to all fugacities below $lambda_c(Delta)$, the uniqueness threshold on the infinite $Delta$-regular tree.
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Let $Phi = (V, mathcal{C})$ be a constraint satisfaction problem on variables $v_1,dots, v_n$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most $[q]$. Suppose that eac
h constraint shares variables with at most $Delta$ constraints and that each constraint is violated with probability at most $p$ (under the product measure on its variables). We show that for $k, q = O(1)$, there is a deterministic, polynomial time algorithm to approximately count the number of satisfying assignments and a randomized, polynomial time algorithm to sample from approximately the uniform distribution on satisfying assignments, provided that [Ccdot q^{3}cdot k cdot p cdot Delta^{7} < 1, quad text{where }C text{ is an absolute constant.}] Previously, a result of this form was known essentially only in the special case when each constraint is violated by exactly one assignment to its variables. For the special case of $k$-CNF formulas, the term $Delta^{7}$ improves the previously best known $Delta^{60}$ for deterministic algorithms [Moitra, J.ACM, 2019] and $Delta^{13}$ for randomized algorithms [Feng et al., arXiv, 2020]. For the special case of properly $q$-coloring $k$-uniform hypergraphs, the term $Delta^{7}$ improves the previously best known $Delta^{14}$ for deterministic algorithms [Guo et al., SICOMP, 2019] and $Delta^{9}$ for randomized algorithms [Feng et al., arXiv, 2020].
We study the problem of sampling an approximately uniformly random satisfying assignment for atomic constraint satisfaction problems i.e. where each constraint is violated by only one assignment to its variables. Let $p$ denote the maximum probabilit
y of violation of any constraint and let $Delta$ denote the maximum degree of the line graph of the constraints. Our main result is a nearly-linear (in the number of variables) time algorithm for this problem, which is valid in a Lovasz local lemma type regime that is considerably less restrictive compared to previous works. In particular, we provide sampling algorithms for the uniform distribution on: (1) $q$-colorings of $k$-uniform hypergraphs with $Delta lesssim q^{(k-4)/3 + o_{q}(1)}.$ The exponent $1/3$ improves the previously best-known $1/7$ in the case $q, Delta = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $1/9$ in the general case [Feng, He, Yin; STOC 2021]. (2) Satisfying assignments of Boolean $k$-CNF formulas with $Delta lesssim 2^{k/5.741}.$ The constant $5.741$ in the exponent improves the previously best-known $7$ in the case $k = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $13$ in the general case [Feng, He, Yin; STOC 2021]. (3) Satisfying assignments of general atomic constraint satisfaction problems with $pcdot Delta^{7.043} lesssim 1.$ The constant $7.043$ improves upon the previously best-known constant of $350$ [Feng, He, Yin; STOC 2021]. At the heart of our analysis is a novel information-percolation type argument for showing the rapid mixing of the Glauber dynamics for a carefully constructed projection of the uniform distribution on satisfying assignments. Notably, there is no natural partial order on the space, and we believe that the techniques developed for the analysis may be of independent interest.
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