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Register a new userApproximate counting and sampling via local central limit theorems
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Added by
Vishesh Jain
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 each 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 probability 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.
We show fully polynomial time randomized approximation schemes (FPRAS) for counting matchings of a given size, or more generally sampling/counting monomer-dimer systems in planar, not-necessarily-bipartite, graphs. While perfect matchings on planar graphs can be counted exactly in polynomial time, counting non-perfect matchings was shown by [Jer87] to be #P-hard, who also raised the question of whether efficient approximate counting is possible. We answer this affirmatively by showing that the multi-site Glauber dynamics on the set of monomers in a monomer-dimer system always mixes rapidly, and that this dynamics can be implemented efficiently on downward-closed families of graphs where counting perfect matchings is tractable. As further applications of our results, we show how to sample efficiently using multi-site Glauber dynamics from partition-constrained strongly Rayleigh distributions, and nonsymmetric determinantal point processes. In order to analyze mixing properties of the multi-site Glauber dynamics, we establish two notions for generating polynomials of discrete set-valued distributions: sector-stability and fractional log-concavity. These notions generalize well-studied properties like real-stability and log-concavity, but unlike them robustly degrade under useful transformations applied to the distribution. We relate these notions to pairwise correlations in the underlying distribution and the notion of spectral independence introduced by [ALO20], providing a new tool for establishing spectral independence based on geometry of polynomials. As a byproduct of our techniques, we show that polynomials avoiding roots in a sector of the complex plane must satisfy what we call fractional log-concavity; this extends a classic result established by [Gar59] who showed homogeneous polynomials that have no roots in a half-plane must be log-concave over the positive orthant.
We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86, JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC. For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csa75]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.
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