TY - UNPB

T1 - Deterministic Approximate Counting for Degree-2 Polynomial Threshold Functions.

AU - De, Anindya

AU - Diakonikolas, Ilias

AU - Servedio, Rocco A.

PY - 2013

Y1 - 2013

N2 - We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-2 polynomial threshold function. Given a degree-2 input polynomial p(x 1 ,…,x n ) and a parameter $\eps > 0$, the algorithm approximates
Pr x∼{−1,1}n [p(x)≥0]
to within an additive $\pm \eps$ in time $\poly(n,2^{\poly(1/\eps)})$. Note that it is NP-hard to determine whether the above probability is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms. This is the first deterministic algorithm for this counting problem in which the running time is polynomial in n for $\eps= o(1)$. For "regular" polynomials p (those in which no individual variable's influence is large compared to the sum of all n variable influences) our algorithm runs in $\poly(n,1/\eps)$ time. The algorithm also runs in $\poly(n,1/\eps)$ time to approximate Pr x∼N(0,1)n [p(x)≥0] to within an additive $\pm \eps$, for any degree-2 polynomial p .
As an application of our counting result, we give a deterministic FPT multiplicative $(1 \pm \eps)$-approximation algorithm to approximate the k -th absolute moment $\E_{x \sim \{-1,1\}^n}[|p(x)^k|]$ of a degree-2 polynomial. The algorithm's running time is of the form $\poly(n) \cdot f(k,1/\eps)$.

AB - We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-2 polynomial threshold function. Given a degree-2 input polynomial p(x 1 ,…,x n ) and a parameter $\eps > 0$, the algorithm approximates
Pr x∼{−1,1}n [p(x)≥0]
to within an additive $\pm \eps$ in time $\poly(n,2^{\poly(1/\eps)})$. Note that it is NP-hard to determine whether the above probability is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms. This is the first deterministic algorithm for this counting problem in which the running time is polynomial in n for $\eps= o(1)$. For "regular" polynomials p (those in which no individual variable's influence is large compared to the sum of all n variable influences) our algorithm runs in $\poly(n,1/\eps)$ time. The algorithm also runs in $\poly(n,1/\eps)$ time to approximate Pr x∼N(0,1)n [p(x)≥0] to within an additive $\pm \eps$, for any degree-2 polynomial p .
As an application of our counting result, we give a deterministic FPT multiplicative $(1 \pm \eps)$-approximation algorithm to approximate the k -th absolute moment $\E_{x \sim \{-1,1\}^n}[|p(x)^k|]$ of a degree-2 polynomial. The algorithm's running time is of the form $\poly(n) \cdot f(k,1/\eps)$.

M3 - Working paper

VL - abs/1311.7105

BT - Deterministic Approximate Counting for Degree-2 Polynomial Threshold Functions.

PB - ArXiv

ER -