9,035 research outputs found

    A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

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    The approximate degree of a Boolean function f ⁣:{1,1}n{1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\} is the least degree of a real polynomial that approximates ff pointwise to error at most 1/31/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function ff with approximate degree dd into a function FF on O(npolylog(n))O(n \cdot \operatorname{polylog}(n)) variables with approximate degree at least D=Ω(n1/3d2/3)D = \Omega(n^{1/3} \cdot d^{2/3}). In particular, if d=n1Ω(1)d= n^{1-\Omega(1)}, then DD is polynomially larger than dd. Moreover, if ff is computed by a polynomial-size Boolean circuit of constant depth, then so is FF. By recursively applying our transformation, for any constant δ>0\delta > 0 we exhibit an AC0^0 function of approximate degree Ω(n1δ)\Omega(n^{1-\delta}). This improves over the best previous lower bound of Ω(n2/3)\Omega(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of nn that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant δ>0\delta > 0, an Ω(n1δ)\Omega(n^{1-\delta}) lower bound on the quantum communication complexity of a function in AC0^0. * A Boolean function ff with approximate degree at least C(f)2o(1)C(f)^{2-o(1)}, where C(f)C(f) is the certificate complexity of ff. This separation is optimal up to the o(1)o(1) term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC0^0.Comment: 40 pages, 1 figur

    Query-Efficient Locally Decodable Codes of Subexponential Length

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    We develop the algebraic theory behind the constructions of Yekhanin (2008) and Efremenko (2009), in an attempt to understand the ``algebraic niceness'' phenomenon in Zm\mathbb{Z}_m. We show that every integer m=pq=2t1m = pq = 2^t -1, where pp, qq and tt are prime, possesses the same good algebraic property as m=511m=511 that allows savings in query complexity. We identify 50 numbers of this form by computer search, which together with 511, are then applied to gain improvements on query complexity via Itoh and Suzuki's composition method. More precisely, we construct a 3r/23^{\lceil r/2\rceil}-query LDC for every positive integer r<104r<104 and a (3/4)512r\left\lfloor (3/4)^{51}\cdot 2^{r}\right\rfloor-query LDC for every integer r104r\geq 104, both of length NrN_{r}, improving the 2r2^r queries used by Efremenko (2009) and 32r23\cdot 2^{r-2} queries used by Itoh and Suzuki (2010). We also obtain new efficient private information retrieval (PIR) schemes from the new query-efficient LDCs.Comment: to appear in Computational Complexit

    Query-to-Communication Lifting for BPP

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    For any nn-bit boolean function ff, we show that the randomized communication complexity of the composed function fgnf\circ g^n, where gg is an index gadget, is characterized by the randomized decision tree complexity of ff. In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.Comment: 21 page

    Parameterized Two-Player Nash Equilibrium

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    We study the computation of Nash equilibria in a two-player normal form game from the perspective of parameterized complexity. Recent results proved hardness for a number of variants, when parameterized by the support size. We complement those results, by identifying three cases in which the problem becomes fixed-parameter tractable. These cases occur in the previously studied settings of sparse games and unbalanced games as well as in the newly considered case of locally bounded treewidth games that generalizes both these two cases

    Tight Size-Degree Bounds for Sums-of-Squares Proofs

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    We exhibit families of 44-CNF formulas over nn variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) dd but require SOS proofs of size nΩ(d)n^{\Omega(d)} for values of d=d(n)d = d(n) from constant all the way up to nδn^{\delta} for some universal constantδ\delta. This shows that the nO(d)n^{O(d)} running time obtained by using the Lasserre semidefinite programming relaxations to find degree-dd SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP\mathsf{NP}-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, M\"uller, and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively
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