Deterministic Approximate Counting for Degree-2 Polynomial Threshold Functions
Servedio, Rocco · 2014
We give a 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(x1, . . . , xn) and a parameter > 0, the algorithm approximates Prx∼{−1,1}n [p(x) ≥ 0] to within an additive ± in time poly(n, 2 ). 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 = 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/ ) time. The algorithm also runs in poly(n, 1/ ) time to approximate Prx∼N(0,1)n [p(x) ≥ 0] to within an additive ± , for any degree-2 polynomial p. As an application of our counting result, we give a deterministic FPT multiplicative (1± )-approximation algorithm to approximate the k-th absolute moment Ex∼{−1,1}n [|p(x)|] of a degree-2 polynomial. The algorithm’s running time is of the form poly(n) · f(k, 1/ ). ∗[email protected]. Research supported by Templeton Foundation Grant 21674 and NSF CCF-1149843. †[email protected]. Some of this work was done while the author was at UC Berkeley supported by a Simons Fellowship. ‡[email protected]. Supported by NSF grants CNS-0716245, CCF-0915929, and CCF-1115703. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 172 (2013)