We develop efficient deterministic algorithms for approximating the fraction of truth assignments that satisfy a disjunctive normal form formula. Although the algorithms themselves are deterministic, their analysis is probabilistic and uses the notion of limited independence between random variables. International Computer Science Institute, 1947 Center Street, Berkeley, California 94704 and Computer Science Department, UC Berkeley, research partially supported by NSF operating grant CCR-9016468 and by grant No. 89-00312 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. y Department of Mathematics, U.C. Berkeley, research partially supported by NSF, research partially done while visiting the International Computer Science Institute ii 1 Introduction Throughout this paper, let F denote a formula in disjunctive normal form (DNF) on n variables with m clauses of length at most t, and let Pr[F ] denote the probability that a random, independent and...
We develop efficient deterministic algorithms for approximating the fraction of truth assignments that satisfy a disjunctive normal form formula. Although the algorithms themselves are deterministic, their analysis is probabilistic and uses the notion of limited independence between random variables. International Computer Science Institute, 1947 Center Street, Berkeley, California 94704 and Computer Science Department, UC Berkeley, research partially supported by NSF operating grant CCR-9016468 and by grant No. 89-00312 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. y Department of Mathematics, U.C. Berkeley, research partially supported by NSF, research partially done while visiting the International Computer Science Institute ii 1 Introduction Throughout this paper, let F denote a formula in disjunctive normal form (DNF) on n variables with m clauses of length at most t, and let Pr[F ] denote the probability that a random, independent and...
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/ ). ∗anindya@math.ias.edu. Research supported by Templeton Foundation Grant 21674 and NSF CCF-1149843. †ilias.d@ed.ac.uk. Some of this work was done while the author was at UC Berkeley supported by a Simons Fellowship. ‡rocco@cs.columbia.edu. Supported by NSF grants CNS-0716245, CCF-0915929, and CCF-1115703. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 172 (2013)
The approximate non-deterministic degree of a Boolean function f, denoted ndegε(f) (written Nε(f) for brevity), is the minimum degree of a real polynomial p such that 0≤∣p(x)∣≤ε whenever f(x)=0, and ∣p(x)∣≥1 whenever f(x)=1. Unlike exact non-deterministic degree, which only requires the polynomial to be nonzero on 1-inputs, this measure enforces a uniform gap: the polynomial must stay close to zero on all 0-inputs and bounded away from zero on all 1-inputs.
The rational degree conjecture, open for over three decades, was recently resolved by Kothari, Kovacs-Deak, Wang, and Yang, who showed that for every total Boolean function f, deg(f)≤O(rdeg(f)3).
In their paper, they explicitly propose a stronger conjecture: that approximate degree is polynomially bounded by Nε(f) and Nε(f) jointly, i.e., for every total Boolean function f and every constant 0<ε<1, deg(f)≤poly(Nε(f),Nε(f)).
This conjecture, if true, would imply a polynomial version of the rational degree result and bring us closer to resolving de Wolf's longstanding non-deterministic degree conjecture.
In this work, we make the first systematic progress on this problem, establishing the conjecture for several broad and natural function classes: monotone and unate functions, functions of bounded alternation number, symmetric functions, k-uniform hypergraph properties, and read-k Disjunctive Normal Form (DNF) formulas.
Abstract. The approximate degree of a Boolean function [Formula: see text] is the minimum degree of a real polynomial [Formula: see text] that approximates [Formula: see text] pointwise: [Formula: see text] for all [Formula: see text]. For every [Formula: see text] we construct conjunctive normal form (CNF) and disjunctive normal form (DNF) formulas of polynomial size with approximate degree [Formula: see text] essentially matching the trivial upper bound of [Formula: see text]. This improves polynomially on previous lower bounds and fully resolves the approximate degree of constant-depth circuits [Formula: see text] a question that has seen extensive research over the past 10 years. Prior to our work, an [Formula: see text] lower bound was known only for [Formula: see text] circuits of depth that grows with [Formula: see text] (M. Bun and J. Thaler, SIAM J. Comput., 49 (2020), pp. FOCS17-59–FOCS17-96). Furthermore, the CNF and DNF formulas that we construct are the simplest possible in that they have constant width. Our proof departs significantly from previous approaches and contributes a novel, number-theoretic method for amplifying approximate degree. Our main result remains valid even for one-sided approximation: for any [Formula: see text], we construct a polynomial-size constant-width CNF formula with one-sided approximate degree [Formula: see text]. We thus obtain the following nearly tight separations: [Formula: see text] versus [Formula: see text] for the one-sided versus two-sided approximate degree of a function and [Formula: see text] versus [Formula: see text] for the one-sided approximate degree of a function [Formula: see text] versus its negation [Formula: see text]. As an application, we essentially settle the communication complexity of [Formula: see text] circuits in the bounded-error quantum model, [Formula: see text]-party number-on-the-forehead randomized model, and [Formula: see text]-party number-on-the-forehead nondeterministic model: we prove that for every [Formula: see text], these models require [Formula: see text], [Formula: see text], and [Formula: see text], respectively, bits of communication even for polynomial-size constant-width CNF formulas. In particular, we show that the multiparty communication class [Formula: see text] can be separated near-optimally from [Formula: see text] and [Formula: see text] by a particularly simple function, a polynomial-size constant-width CNF formula.
We consider the problem of approximating a Boolean function f : f0; 1g n ! f0; 1g by the sign of an integer polynomial p of degree k. For us, a polynomial p(x) predicts the value of f(x) if, whenever p(x) 0, f(x) = 1, and whenever p(x) ! 0, f(x) = 0. A low-degree polynomial p is a good approximator for f if it predicts f at almost all points. Given a positive integer k, and a Boolean function f , we ask, "how good is the best degree k approximation to f?" We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the case f is parity. Minsky and Papert [10] proved that a perceptron can not compute parity; our bounds indicate exactly how well Yale University, Dept. of Computer Science, P.O. Box 208285, New Haven CT 06520-8285. y Email: aspnes-james@cs.yale.edu. z Email: beigel-richard@cs.yale.edu. Supported in part by NSF grants CCR-8808949 and CCR-8958528. x Carnegie-Mellon University, Schoo...
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) over Rn and a parameter ε > 0, our algorithm approximates Pr [EQUATION] to within an additive ±ε in time Od,ε(1) · poly(nd). (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 nd, the number of coefficients of a degree-d PTF) is a fixed polynomial. The fastest previous algorithm for this problem [Kan12b], based on constructions of unconditional pseudorandom generators for degree-d PTFs, runs in time [EQUATION] for all c > 0.
A garbling scheme is used to garble a circuit C and an input x in a way that reveals the output C(x) but hides everything else. In many settings, the circuit can be garbled off-line without strict efficiency constraints, but the input must be garbled very efficiently on-line, with much lower complexity than evaluating the circuit. Yao’s scheme has essentially optimal on-line complexity, but only achieves selective security, where the adversary must choose the input x prior to seeing the garbled circuit. It has remained an open problem to achieve adaptive security, where the adversary can choose x after seeing the garbled circuit, while preserving on-line efficiency. In this work, we modify Yao’s scheme in a way that allows us to prove adaptive security under one-way functions. As our main instantiation, we get a scheme where the on-line complexity is only proportional to the width w of the circuit, which corresponds to the space complexity of the computation, but is independent of the circuit’s depth d. Alternately, we can also get an instantiation where the on-line complexity is only proportional to the input/output size and the depth d of the circuit but independent of its width w, albeit in this case we incur a 2 security loss in our reduction. More broadly, we relate the on-line complexity of adaptively secure garbling schemes in our framework to a certain type of pebble complexity of the circuit. As our main tool, of independent interest, we develop a new notion of somewhere equivocal encryption, which allows us to efficiently equivocate on a small subset of the message bits. ∗This work was done in part while some of the authors were visiting the Simons Institute for the Theory of Computing, supported by the Simons Foundation and by the DIMACS/Simons Collaboration in Cryptography through NSF grant CNS1523467. †University of Pennsylvania. Department of Computer Science. fbrett@cis.upenn.edu ‡Northeastern University. Department of Computer Science. zahra@ccs.neu.edu §University of California, Los Angeles. Department of Computer Science and Mathematics. rafail@cs.ucla.edu Research supported in part by NSF grants 09165174, 1065276, 1118126 and 1136174, US-Israel BSF grant 2008411, OKAWA Foundation Research Award, IBM Faculty Research Award, Xerox Faculty Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. This material is based upon work supported in part by DARPA Safeware program. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. ¶Boston University and Northeastern University. Department of Computer Science. scafuro@bu.edu Research supported by NSF grants 1012798. ‖Northeastern University. Department of Computer Science. wichs@ccs.neu.edu. Research supported by NSF grants CNS-1347350, CNS-1314722, CNS-1413964.