TR13-173 Authors: Anindya De, Rocco Servedio

Publication: 3rd December 2013 22:24

Downloads: 2511

Keywords:

We give a deterministic algorithm for

approximately counting satisfying assignments of a degree-$d$ polynomial threshold function

(PTF).

Given a degree-$d$ input polynomial $p(x_1,\dots,x_n)$ over $\mathbb{R}^n$

and a parameter $\epsilon > 0$, our algorithm approximates

$

\mathbf{P}_{x \sim \{-1,1\}^n}[p(x) \geq 0]

$

to within an additive $\pm \epsilon$ in time $O_{d,\epsilon}(1)\cdot \mathop{poly}(n^d)$.

(Since it is NP-hard to determine whether the above probability

is nonzero, any sort of efficient multiplicative approximation is

almost certainly impossible even for randomized algorithms.)

Note that the running time of our algorithm (as a function of $n^d$,

the number of coefficients of a degree-$d$ PTF)

is a \emph{fixed} polynomial. The fastest previous algorithm for

this problem (due to Kane), based on constructions of

unconditional pseudorandom generators for degree-$d$ PTFs, runs in time

$n^{O_{d,c}(1) \cdot \epsilon^{-c}}$ for all $c > 0$.

The key novel contributions of this work are:

A new multivariate central limit theorem, proved using tools

from Malliavin calculus and Stein's Method. This new CLT shows that any

collection of Gaussian polynomials with small eigenvalues must have a

joint distribution which is very close to a multidimensional Gaussian

distribution.

A new decomposition of low-degree multilinear polynomials over Gaussian

inputs. Roughly speaking we show that (up to some small error)

any such polynomial can be decomposed

into a bounded number of multilinear polynomials all of which have

extremely small eigenvalues.

We use these new ingredients to give a deterministic algorithm

for a Gaussian-space version of the approximate counting problem,

and then employ standard techniques for working with

low-degree PTFs (invariance principles and regularity lemmas)

to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian

version.

As an application of our result, we give the first deterministic fixed-parameter tractable

algorithm for the following moment approximation problem: given a degree-$d$ polynomial $p(x_1,\dots,x_n)$ over $\{-1,1\}^n$, a positive integer

$k$ and an error parameter $\epsilon$, output a $(1\pm \epsilon)$-multiplicatively

accurate estimate to $\mathbf{E}_{x \sim \{-1,1\}^n}[|p(x)|^k].$ Our algorithm

runs in time $O_{d,\epsilon,k}(1) \cdot \mathop{poly}(n^d).$