Computing approximate roots of monotone functions

IF 1.8 2区数学 Q1 MATHEMATICS

Journal of Complexity Pub Date : 2025-01-28 DOI:10.1016/j.jco.2025.101930

Alexandros Hollender , Chester Lawrence, Erel Segal-Halevi

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引用次数: 0

Abstract

We are given a value-oracle for a d-dimensional function f that satisfies the conditions of Miranda's theorem, and therefore has a root. Our goal is to compute an approximate root using a number of evaluations that is polynomial in the number of accuracy digits. For

d = 1

this is always possible using the bisection method, but for

d \geq 2

this is impossible in general.

We show that, if

d = 2

and f satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. The same holds if

d \geq 3

and f satisfies some particular

d^{2} - d

monotonicity conditions. We show that, if

d = 2

and f satisfies a single monotonicity condition, then the number of required evaluations is polynomial in the accuracy. The same holds if

d \geq 3

and f satisfies some particular

d^{2} - d

monotonicity conditions. In contrast, if even two of these monotonicity conditions are missing, then the required number of evaluations might be exponential.

As an example application, we show that approximate roots of monotone functions can be used for approximate envy-free cake-cutting.

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来源期刊

Journal of Complexity 工程技术-计算机：理论方法

CiteScore

3.10

自引率

17.60%

发文量

审稿时长

>12 weeks

期刊介绍： The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.