891 research outputs found

    Trade-Offs Between Size and Degree in Polynomial Calculus

    Get PDF
    Building on [Clegg et al. \u2796], [Impagliazzo et al. \u2799] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(?(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen \u2716], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary

    Polynomial calculus space and resolution width

    Get PDF
    We show that if a k-CNF requires width w to refute in resolution, then it requires space square root of √ω to refute in polynomial calculus, where the space of a polynomial calculus refutation is the number of monomials that must be kept in memory when working through the proof. This is the first analogue, in polynomial calculus, of Atserias and Dalmau's result lower-bounding clause space in resolution by resolution width. As a by-product of our new approach to space lower bounds we give a simple proof of Bonacina's recent result that total space in resolution (the total number of variable occurrences that must be kept in memory) is lower-bounded by the width squared. As corollaries of the main result we obtain some new lower bounds on the PCR space needed to refute specific formulas, as well as partial answers to some open problems about relations between space, size, and degree for polynomial calculus

    Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

    Get PDF
    We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose degree is of the order of the square root of n log s plus k. A similar statement holds for the more general Positivstellensatz (PS) proofs. This establishes size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof systems for the LP and SDP hierarchies of LovĂĄsz and Schrijver. As a corollary to this, and to the known degree lower bounds, we get optimal integrality gaps for exponential size SOS proofs for sparse random instances of the standard NP-hard constraint optimization problems. We also get exponential size SOS lower bounds for Tseitin and Knapsack formulas. The proof of our main result relies on a zero-gap duality theorem for pre-ordered vector spaces that admit an order unit, whose specialization to PS and SOS may be of independent interest.Peer ReviewedPostprint (published version

    Some subsystems of constant-depth Frege with parity

    Get PDF
    We consider three relatively strong families of subsystems of AC0[2]-Frege proof systems, i.e., propositional proof systems using constant-depth formulas with an additional parity connective, for which exponential lower bounds on proof size are known. In order of increasing strength, the subsystems are (i) constant-depth proof systems with parity axioms and the (ii) treelike and (iii) daglike versions of systems introduced by Krajíček which we call PKcd(⊕). In a PKcd(⊕)-proof, lines are disjunctions (cedents) in which all disjuncts have depth at most d, parities can only appear as the outermost connectives of disjuncts, and all but c disjuncts contain no parity connective at all. We prove that treelike PKO(1)O(1)(⊕) is quasipolynomially but not polynomially equivalent to constant-depth systems with parity axioms. We also verify that the technique for separating parity axioms from parity connectives due to Impagliazzo and Segerlind can be adapted to give a superpolynomial separation between daglike PKO(1)O(1)(⊕) and AC0[2]-Frege; the technique is inherently unable to prove superquasipolynomial separations. We also study proof systems related to the system Res-Lin introduced by Itsykson and Sokolov. We prove that an extension of treelike Res-Lin is polynomially simulated by a system related to daglike PKO(1)O(1)(⊕), and obtain an exponential lower bound for this system.Peer ReviewedPostprint (author's final draft

    Total Space in Resolution Is at Least Width Squared

    Get PDF
    Given an unsatisfiable k-CNF formula phi we consider two complexity measures in Resolution: width and total space. The width is the minimal W such that there exists a Resolution refutation of phi with clauses of at most W literals. The total space is the minimal size T of a memory used to write down a Resolution refutation of phi where the size of the memory is measured as the total number of literals it can contain. We prove that T = Omega((W - k)^2)

    Online Computation with Untrusted Advice

    Get PDF
    The advice model of online computation captures the setting in which the online algorithm is given some partial information concerning the request sequence. This paradigm allows to establish tradeoffs between the amount of this additional information and the performance of the online algorithm. However, unlike real life in which advice is a recommendation that we can choose to follow or to ignore based on trustworthiness, in the current advice model, the online algorithm treats it as infallible. This means that if the advice is corrupt or, worse, if it comes from a malicious source, the algorithm may perform poorly. In this work, we study online computation in a setting in which the advice is provided by an untrusted source. Our objective is to quantify the impact of untrusted advice so as to design and analyze online algorithms that are robust and perform well even when the advice is generated in a malicious, adversarial manner. To this end, we focus on well- studied online problems such as ski rental, online bidding, bin packing, and list update. For ski-rental and online bidding, we show how to obtain algorithms that are Pareto-optimal with respect to the competitive ratios achieved; this improves upon the framework of Purohit et al. [NeurIPS 2018] in which Pareto-optimality is not necessarily guaranteed. For bin packing and list update, we give online algorithms with worst-case tradeoffs in their competitiveness, depending on whether the advice is trusted or not; this is motivated by work of Lykouris and Vassilvitskii [ICML 2018] on the paging problem, but in which the competitiveness depends on the reliability of the advice. Furthermore, we demonstrate how to prove lower bounds, within this model, on the tradeoff between the number of advice bits and the competitiveness of any online algorithm. Last, we study the effect of randomization: here we show that for ski-rental there is a randomized algorithm that Pareto-dominates any deterministic algorithm with advice of any size. We also show that a single random bit is not always inferior to a single advice bit, as it happens in the standard model

    Optimal approximation of piecewise smooth functions using deep ReLU neural networks

    Full text link
    We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in L2L^2. As a model class, we consider the set Eβ(Rd)\mathcal{E}^\beta (\mathbb R^d) of possibly discontinuous piecewise CβC^\beta functions f:[−1/2,1/2]d→Rf : [-1/2, 1/2]^d \to \mathbb R, where the different smooth regions of ff are separated by CβC^\beta hypersurfaces. For dimension d≥2d \geq 2, regularity β>0\beta > 0, and accuracy ε>0\varepsilon > 0, we construct artificial neural networks with ReLU activation function that approximate functions from Eβ(Rd)\mathcal{E}^\beta(\mathbb R^d) up to L2L^2 error of ε\varepsilon. The constructed networks have a fixed number of layers, depending only on dd and β\beta, and they have O(ε−2(d−1)/β)O(\varepsilon^{-2(d-1)/\beta}) many nonzero weights, which we prove to be optimal. In addition to the optimality in terms of the number of weights, we show that in order to achieve the optimal approximation rate, one needs ReLU networks of a certain depth. Precisely, for piecewise Cβ(Rd)C^\beta(\mathbb R^d) functions, this minimal depth is given---up to a multiplicative constant---by β/d\beta/d. Up to a log factor, our constructed networks match this bound. This partly explains the benefits of depth for ReLU networks by showing that deep networks are necessary to achieve efficient approximation of (piecewise) smooth functions. Finally, we analyze approximation in high-dimensional spaces where the function ff to be approximated can be factorized into a smooth dimension reducing feature map τ\tau and classifier function gg---defined on a low-dimensional feature space---as f=g∘τf = g \circ \tau. We show that in this case the approximation rate depends only on the dimension of the feature space and not the input dimension.Comment: Generalized some estimates to LpL^p norms for $0<p<\infty
    • …
    corecore