pinecone/core-2020-05-10-deduplication huggingface dataset:
On NP-Completeness for Linear Machines

AbstractWe consider linear and scalar versions of the Blum–Shub–Smale model of computation over the reals. The permitted computing operations of linear machines are addition and multiplication by constants. The scalar machines can only multiply by constants. The size of an input is its dimension, and the cost of any instruction is one. For each of these structures we consider DNP and NP, the corresponding complexity classes with respect to digital nondeterminism and standard real nondeterminism, respectively. We give DNP- and NP-complete problems for linear and real scalar machines. On the other hand, we show that the NP-class restricted to scalar machines over the integers with equality-tests does not own a complete problem
Real Data—Integer Solution Problems within the Blum–Shub–Smale Computational Model

AbstractThis paper presents results connected with the theory of computation over the reals, developed recently by Blum, Shub, and Smale. The subjects of our investigation are variations of the well known Integer Linear Programming Problem and the Linear Diophantine Equation, where the coefficients are real numbers and we look for an integer solution. We show some results concerning the solvability and complexity of these problems, and we develop algorithms for their solution in the presence of additional information
Some Complexity Results for Polynomial Ideals

AbstractIn this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w+ 1)-tupleP= (f,g1,g2, …gwwherefand thegiare multivariate polynomials, and the problem is to determine whetherfis in the ideal generated by thegi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases
Shape from Shadows A Hilbert Space Setting

AbstractWe study the problem of recovering a surface from the shadows it casts on itself when lighted by the sun at various times of the day. Shadows can create both linear and nonlinear information. We will show how to incorporate both types of information in the solution. The problem is formulated and solved in a Hilbert space setting and the spline algorithm interpolating the data that result from the shadows is constructed. This algorithm is optimal in terms of the approximation error and has low cost. We furthermore derive optimal information for this problem
Approximability of the Ground State Problem for Certain Ising Spin Glasses

AbstractWe consider polynomial time algorithms for finding approximate solutions to the ground state problem for the following three-dimensional case of an Ising spin glass: 2nspins are arranged on a two-level grid with at mostnγvertical interactions (0 ≤ γ ≤ 1). The main results are:1.Let12≤ γ < 1. There is an approximate polynomial time algorithm with absolute error less thannγfor alln; there exists a constant β > 0 such that every approximate polynomial time algorithm has absolute error greater than βnγinfinitely often, unlessP=NP.2.Let γ = 1. There is an approximate polynomial time algorithm with absolute error less thann/lgn; there exists a numberk> 1 such that every approximate polynomial time algorithm has absolute error greater thann/(lgn)kinfinitely often iffNP⊈ ∩ε > 0DTIME(2nε)
Recognizing Global Occurrence of Local Properties

AbstractLet P be a graph property. Fork≥ 1, a graphGhas propertyPkiff every inducedk-vertex subgraph ofGhas P. For a graphGwe denote byNPk(G) the number of inducedk-vertex subgraphs ofGhaving P. A property is calledspanningif it does not hold for graphs that contain isolated vertices. A property is calledconnectedif it does not hold for graphs with more than one connected component. Many familiar graph properties are spanning or connected. We also define the notion ofsimpleproperties which also applies to many well-known monotone graph properties. A property P is recursive if one can determine if a graphGonnvertices has P in timeO(fP(n)) wherefP(n) is some recursive function ofn. We consider only recursive properties. Our main results are the following.•If P is spanning andk≥ 1 is fixed, deciding whether a graphG= (V,E) hasPkcan be done inO(V+E) time.•If P is spanning,fP(n) = O(2n3), andk=O((logn/log logn)1/3), deciding whetherGhasPkcan be done in polynomial time. Furthermore, if P is a monotone-increasing simple property withfP(n) =O(2n2) (Hamiltonicity, perfect-matching, ands-connectivity are just a few examples of such properties) andk=O(logn/log logn), deciding whetherGhasPkcan be done in polynomial time.•Ifk≥ 1 andd≥ 1 are fixed, and P is either a connected property (Hamiltonicity is an example of such a property) or a monotone-decreasing infinitely-simple property (perfect-matching of independent vertices and the Hamiltonian hole are examples of such properties) computingNPk(G) for graphsGwith Δ(G) ≤dcan be done in linear time.•If P is an NP-Hard monotone property and ε > 0 is fixed, thenP⌊nε⌋is also NP-Hard. The monotonicity is required as there are NP-Hard properties wherePkis easy whenk<n
Linear Complexity Profiles: Hausdorff Dimensions for Almost Perfect Profiles and Measures for General Profiles

AbstractStream ciphers usually employ some sort of pseudorandomly generated bit strings to be added to the plaintext. The cryptographic properties of such a sequenceacan be stated in terms of the so-called linear complexity profile (l.c.p.),La(t),t∈ N. If the l.c.p. isLa(t) =t/2 +O(1), it is called (almost)perfect. This paper examines first those subsets Ad(q)of Fq∞where for fixedd∈ N the l.c.p. satisfies |2 ·La(t) −t| ≤dfor allt∈ N. It turns out that (after suitably mapping Ad(q)on [0, 1] ⊂ R) the Hausdorff dimension is1+logqϕd(q)2where ϕd(q)is the largest real root ofxd= (q− 1) · ∑i=0d−1xi. The second part deals with nondecreasing boundsd: N → N. Sinced(t) → ∞ ast→ ∞ always leads to a Hausdorff dimension 1, here we consider the measure of the set Ad(q)
Tractability of Tensor Product Linear Operators

AbstractThis paper deals with the worst case setting for approximating multivariate tensor product linear operators defined over Hilbert spaces. Approximations are obtained by using a number of linear functionals from a given class of information. We consider the three classes of information: the class of all linear functionals, the Fourier class of inner products with respect to given orthonormal elements, and the standard class of function values. We wish to determine which problems are tractable and which are strongly tractable. The complete analysis is provided for approximating operators of rank two or more. The problem of approximating linear functionals is fully analyzed in the first two classes of information. For the third class of standard information we show that the possibilities are very rich. We prove that tractability of linear functionals depends on the given space of functions. For some spaces all nontrivial normed linear functionals are intractable, whereas for other spaces all linear functionals are tractable. In “typical” function spaces, some linear functionals are tractable and some others are not
On the Randomized Error of Polynomial Methods for Eigenvector and Eigenvalue Estimates

AbstractIn this paper we consider the problem of estimating the largest eigenvalue and the corresponding eigenvector of a symmetric matrix. In particular, we consider iterative methods, such as the power method and the Lanczos method. These methods need a starting vector which is usually chosen randomly. We analyze the behavior of these methods when the initial vector is chosen with uniform distribution over the unitn-dimensional sphere. We extend and generalize the results reported earlier. In particular, we give upper and lower bounds on the Lpnorm of the randomized error, and we improve previously known bounds with a detailed analysis of the role of the multiplicity of the largest eigenvalue
The Complexity of Indefinite Elliptic Problems with Noisy Data

AbstractWe study the complexity of second-order indefinite elliptic problems −div(a∇u) +bu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain Ω, the error being measured in theH1(Ω)-norm. The problem elementsfbelong to the unit ball ofWr,p, (Ω), wherep∈ [2, ∞] andr>d/p. Information consists of (possibly adaptive) noisy evaluations off,a, orb(or their derivatives). The absolute error in each noisy evaluation is at most δ. We find that thenth minimal radius for this problem is proportional ton−r/d+ δ and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the ε-complexity (minimal cost of calculating an ε-approximation) for this problem, said bounds depending on the costc(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation isc(δ) = δ−s(fors> 0), then the complexity is proportional to (1/ε)d/r+s
On the Value of Partial Information for Learning from Examples

AbstractThe PAC model of learning and its extension to real valued function classes provides a well-accepted theoretical framework for representing the problem of learning a target functiong(x) using a random sample {(xi,g(xi))}i=1m. Based on the uniform strong law of large numbers the PAC model establishes the sample complexity, i.e., the sample sizemwhich is sufficient for accurately estimating the target function to within high confidence. Often, in addition to a random sample, some form of prior knowledge is available about the target. It is intuitive that increasing the amount of information should have the same effect on the error as increasing the sample size. But quantitatively how does the rate of error with respect to increasing information compare to the rate of error with increasing sample size? To answer this we consider a new approach based on a combination of information-based complexity of Traubet al.and Vapnik–Chervonenkis (VC) theory. In contrast to VC-theory where function classes of finite pseudo-dimension are used only for statistical-based estimation, we let such classes play a dual role of functional estimation as well as approximation. This is captured in a newly introduced quantity, ρd(F), which represents a nonlinear width of a function class F. We then extend the notion of thenth minimal radius of information and define a quantityIn,d(F) which measures the minimal approximation error of the worst-case targetg∈ F by the family of function classes having pseudo-dimensiondgiven partial information ongconsisting of values taken bynlinear operators. The error rates are calculated which leads to a quantitative notion of the value of partial information for the paradigm of learning from examples
When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals?

AbstractRecently, quasi-Monte Carlo algorithms have been successfully used for multivariate integration of high dimensiond, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasi-Monte Carlo algorithms does not explain this phenomenon. This paper presents a partial answer to why quasi-Monte Carlo algorithms can work well for arbitrarily larged. It is done by identifying classes of functions for which the effect of the dimensiondis negligible. These areweightedclasses in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimalworst caseerror of quasi-Monte Carlo algorithms does not depend on the dimensiondiff the sum of the weights is finite. We also prove that the minimal number of function values in the worst case setting needed to reduce the initial error by ε is bounded byCε−p, where the exponentp∈ [1, 2], andCdepends exponentially on the sum of weights. Hence, the relatively small sum of the weights makes some quasi-Monte Carlo algorithms strongly tractable. We show in a nonconstructive way that many quasi-Monte Carlo algorithms are strongly tractable. Even random selection of sample points (done once for the whole weighted class of functions and then the worst case error is established for that particular selection, in contrast to Monte Carlo where random selection of sample points is carried out for a fixed function) leads to strong tractable quasi-Monte Carlo algorithms. In this case the minimal number of function values in theworst casesetting is of order ε−pwith the exponentp= 2. The deterministic construction of strongly tractable quasi-Monte Carlo algorithms as well as the minimal exponentpis open
Optimal Integration of Lipschitz Functions with a Gaussian Weight

AbstractWe study optimal integration over the infinite interval (−∞, +∞) for a Gaussian weight with variance σ. We consider functions satisfying a Lipschitz condition with constantL. We characterize the optimal information by a system of nonlinear equations, and show that (asymptotically) the solution is defined by the quantiles of the Gaussian weight with double variance 2σ. We provide an optimal algorithm and prove that fornoptimal sample points the worst case error of the optimal algorithm is asymptotically equal toπσ/2L/n. Finally, we show that the worst case error for the set of sampling points defined by the zeros of Hermite polynomials is quadratically worse than for optimal sample points
A Refined Model of Computation for Continuous Problems

AbstractThe real-number model of computation is used in computational geometry, in the approach suggested by Blum, Shub, and Smale and in information based complexity. In this paper we propose a refinement of this model, the TTE-model of computation. In contrast to the real-number model, which is unrealistic or at least too optimistic, the TTE-model is very realistic; i.e., for TTE-algorithms there are digital computers, which behave exactly the same way as predicted by the theoretical model. We start with a detailed discussion of some objections to the real-number model. We introduce the refined model by adding the condition “every partial input or output information of an algorithm is finite” to the assumptions of the IBC-model of computation. First, we explain computability and computational complexity in TTE for the simple case of real functions. Then we apply the refined model to two typical IBC-problems, integration and zero-finding on the spaceC[0; 1] of continuous real functions. We study computability and computational complexity and compare TTE-results with IBC-results. Finally, we briefly discuss the computation of discontinuous problems. This paper does not present new technical results but should be considered as a fresh impetus to reflect on models of computation for numerical analysis and as an invitation to try out the TTE-model of computation in information based complexity
Linear Algorithms with Finite Deviation

AbstractWe discuss the computation ofI[f], whereIis a positive functional onC[a,b]. Any algorithm starts with some input, in typical situations this input consists of function values (f(x1), …,f(xn)) and the more global informationf∈ K, where K ⊂C[a,b] is a given set. The algorithm which makes the fullest use of the input is called the strongly optimal algorithm. We shall show that in many cases of practical interest these algorithms are not linear in (f(x1), …,f(xn)), in sharp contrast to the better known optimal algorithms. A weakening of the notion of strong optimality leads to algorithms with finite deviation, for these we obtain almost the same results. Furthermore, some examples of linear algorithms with small deviation are given
Monte Carlo Complexity of Global Solution of Integral Equations

AbstractThe problem of the global solution of Fredholm integral equations is studied. This means that one seeks to approximate the full solution function (as opposed to the local problem, where only the value of the solution in a single point or a functional of the solution is sought). The Monte Carlo complexity, i.e., the complexity of the stochastic solution of this problem, is analyzed. The framework for this analysis is provided by information-based complexity theory. The investigations complement previous ones on the stochastic complexity of the local solution and on deterministic complexity of both local and global solutions. The results show that even in the global case Monte Carlo algorithms can perform better than deterministic ones, although the difference is not as large as in the local case
On A Scheduling Problem of Time Deteriorating Jobs

AbstractWe consider a scheduling problem with a single machine and a set of jobs which have to be processed sequentially. While waiting for processing, jobs may deteriorate, causing the processing requirement of each job to grow after a fixed waiting timet0. We prove that the problem of minimizing the makespan—completion time for all jobs—is NP-hard. Next we consider the problem for a natural special case where the job requirement grows linearly at a job-specific rate aftert0. We develop a fully polynomial time approximation scheme for the problem in this case. We also give further NP-hardness results, and a polynomial time algorithm for the case where the job-specific rate is proportional to the initial processing requirement of each job
Lyapunov Exponents versus Expansivity and Sensitivity in Cellular Automata

AbstractWe establish a connection between the theory of Lyapunov exponents and the properties of expansivity and sensitivity to initial conditions for a particular class of discrete time dynamical systems; cellular automata (CA). The main contribution of this paper is the proof that all expansive cellular automata have positive Lyapunov exponents for almost all the phase space configurations. In addition, we provide an elementary proof of the non-existence of expansive CA in any dimension greater than 1. In the second part of this paper we prove that expansivity in dimension greater than 1 can be recovered by restricting the phase space to asuitablesubset of the whole space. To this extent we describe a 2-dimensional CA which is expansive over adense uncountablesubset of the whole phase space. Finally, we highlight the different behavior of expansive and sensitive CA for what concerns the speed at which perturbations propagate
δ-Uniform BSS Machines

AbstractAδ-uniform BSS machine is a standard BSS machine which does not rely on exact equality tests. We prove that, for any real closed archimedean fieldR, a set isδ-uniformly semi-decidable iff it is open and semi-decidable by a BSS machine which is locally time bounded; we also prove that the local time bound condition is nontrivial. This entails a number of results about BSS machines, in particular the existence of decidable sets whose interior (closure) is not even semi-decidable without adding constants. Finally, we show that the sets semi-decidable by Turing machines are the sets semi-decidable byδ-uniform machines with coefficients inQorT, the field of Turing computable numbers
Fast Rectangular Matrix Multiplication and Applications

AbstractFirst we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of ann×nmatrix by ann×n2matrix in arithmetic timeO(nω),ω=3.333953…, which is less by 0.041 than the previous record 3.375477…. Then we present fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimate the asymptotic running time as a function of the dimensions, and optimize the exponents of the complexity estimates. For a large class of input matrix pairs, we improve the known exponents. Finally we show three applications of our results: (a)we decrease from 2.851 to 2.837 the known exponent of the work bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of ann×nmatrix, as well as for the solution to a nonsingular linear system ofnequations, (b)we asymptotically accelerate the known sequential algorithms for the univariate polynomial composition modxn, yielding the complexity boundO(n1.667) versus the old record ofO(n1.688), and for the univariate polynomial factorization over a finite field, and (c)we improve slightly the known complexity estimates for computing basic solutions to the linear programming problem withmconstraints andnvariables
On the Complexity of Self-Validating Numerical Integration and Approximation of Functions with Singularities

AbstractWe consider the complexity of numerical integration and piecewise polynomial at approximation of bounded functions from a subclass ofCk([a,b]\Z), whereZis a finite subset of [a,b]. Using only function values or values of derivatives, we usually cannot guarantee that the costs for obtaining an error less thanεare bounded byO(ε−1/k) and we may have much higher costs. The situation changes if we also allow “realistic” estimates of ranges of functions or derivatives on intervals as observations. A very simple algorithm now yields an error less thanεwithO(ε−1/k)-costs and an analogous result is also obtained for uniform approximation with piecewise polynomials. In a practical implementation, estimation of ranges may be done efficiently with interval arithmetic and automatic differentiation. The cost for each such evaluation (also of ranges of derivatives) is bounded by a constant times the cost for a function evaluation. The mentioned techniques reduce the class of integrands, but still allow numerical integration of functions from a wide class withO(ε−1/k) arithmetical operations and guaranteed precisionε
Relaxation of Product Markov Chains on Product Spaces

AbstractWe study the relaxation time of product-type Markov chains approaching a product distribution. We bound the approach to stationarity for such Markov chains in terms of the mixing times of the component Markov chains. In cases where the component mixing times differ considerably we propose an optimized visiting scheme which makes such product-type Markov chains comparable to Gibbs-type samplers. We conclude the paper by discussing the relaxation of Metropolis-type samplers for separable energy functions
On the Asymptotic Average Number of Efficient Vertices in Multiple Objective Linear Programming

AbstractLeta1,…,am,c1,…,ckbe independent random points in Rnthat are identically distributed spherically symmetrical in Rnand letX≔{x∈Rn|aTix⩽1,i=1,…,m} be the associated random polyhedron form⩾n⩾2. We consider multiple objective linear programming problems maxx∈XcT1x, maxx∈XcT2x,…,maxx∈XcTkxwith 1⩽k⩽n. For distributions with algebraically decreasing tail in the unit ball, we investigate the asymptotic expected number of vertices in the efficient frontier ofXwith respect toc1,…,ckfor fixedn,kandm→∞. This expected number of efficient vertices is the most significant indicator for the average-case complexity of the multiple objective linear programming problem