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On the complexity of computing mixed volumes

SIAM JOURNAL ON COMPUTING 1998年第2期27卷 356-400页

作者： Dyer, M Gritzmann, P Hufnagel, A Univ Leeds Sch Comp Studies Leeds LS2 9JT W Yorkshire England Tech Univ Munich Ctr Math Sci D-80290 Munich Germany

This paper gives various (positive and negative) results on the complexity of the problem of computing and approximating mixed volumes of polytopes and more general convex bodies in arbitrary dimension. On the negative side, we present several # P-hardness results that focus on the difference of computing mixed volumes versus computing the volume of polytopes. We show that computing the volume of zonotopes is # P-hard (while each corresponding mixed volume can be computed easily) but also give examples showing that computing mixed volumes is hard even when computing the volume is easy. On the positive side, we derive a randomized algorithm for computing the mixed volumes [GRAPHICS] of well-presented convex bodies K(1),...,K(s), where m(1),...,m(s) is an element of N(0) and m(1) greater than or equal to n - psi(n) with psi(n) = o(log n/log log n). The algorithm is an interpolation method based on polynomial-time randomized log algorithms for computing the volume of convex bodies. This paper concludes with applications of our results to various problems in discrete mathematics, combinatorics, computational convexity, algebraic geometry, geometry of numbers, and operations research.

关键词： computational convexity volume mixed volumes convex body polytope zonotope parallelotope computation approximation computational complexity deterministic algorithm randomized algorithm polynomial-time algorithm NP-hardness #P-hardness permanent determinant problems lattice point enumerator partial order Newton polytope polynomial equations

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A new algorithm that generates the image of the attractor of a generalized iterated function system

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NUMERICAL algorithmS 2020年第4期83卷 1399-1413页

作者： Miculescu, Radu Mihail, Alexandru Urziceanu, Silviu-Aurelian Transilvania Univ Brasov Fac Math & Comp Sci Iuliu Maniu St50 Brasov 500091 Romania Univ Bucharest Fac Math & Comp Sci Acad St 14 Bucharest 010014 Romania Univ Pitesti Fac Math & Comp Sci Targul Din Vale 1 Pitesti 110040 Arges Romania

We provide a new algorithm (called the grid algorithm) designed to generate the image of the attractor of a generalized iterated function system on a finite dimensional space and we compare it with the deterministic algorithm regarding generalized iterated function systems presented by Jaros et al. (Numer. algorithms 73, 477-499, 2016).

关键词： Generalized infinite iterated function system (GIFS) Attractor deterministic algorithm Grid algorithm

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Global optimization algorithm for a generalized linear multiplicative programming

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2012年第1-2期40卷 551-568页

作者： Jiao, Hongwei Liu, Sanyang Chen, Yongqiang Xidian Univ Dept Math Sci Xian 710071 Peoples R China Henan Inst Sci & Technol Dept Math Xinxiang 453003 Peoples R China Henan Normal Univ Coll Math & Informat Xinxiang 453007 Peoples R China

This article present a branch and bound algorithm for globally solving generalized linear multiplicative programming problems with coefficients. The main computation involve solving a sequence of linear relaxation programming problems, and the algorithm economizes the required computations by conducting the branch and bound search in R-p, rather than in R-n, where p is the number of rank and n is the dimension of decision variables. The proposed algorithm will be convergent to the global optimal solution by means of the subsequent solutions of the series of linear relaxation programming problems. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.

关键词： Multiplicative programming Global optimization Branch and bound algorithm deterministic algorithm

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Values of rational functions in small subgroups of finite fields and the identity testing problem from powers

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INTERNATIONAL JOURNAL OF NUMBER THEORY 2020年第2期16卷 219-231页

作者： Merai, Laszlo Austrian Acad Sci Johann Radon Inst Computat & Appl Math Altenberger Str 69 A-4040 Linz Austria

Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field F-qn considered as a linear space over a subfield F-q. We apply this to the recently introduced algorithmic problem of identity testing of "hidden" polynomials f and g over a high degree extension of a finite field, given oracle access to f(x)(e) and g(x)(e).

关键词： Identity testing from powers hidden polynomial power Nullstellensatz deterministic algorithm

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Discussion of "The power of monitoring: how to make the most of a contaminated multivariate sample"

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STATISTICAL METHODS AND APPLICATIONS 2018年第4期27卷 589-594页

作者： Raymaekers, Jakob Rousseeuw, Peter J. Vranckx, Iwein Univ Leuven Dept Math Celestijnenlaan 200b B-3001 Louvain Belgium

In this comment on the discussion paper The power of monitoring: how to make the most of a contaminated multivariate sample by A. Cerioli, M. Riani, A. Atkinson and A. Corbellini, we describe how the hard rejection property of the MCD method can be mimicked by an S-estimator with appropriate rho-function. We also point the reader to fast and deterministic algorithms for the MCD, S- and MM-estimators that are specifically suited for monitoring experiments. They were made available a few years ago and successfully used for monitoring in our papers. Finally, the question is raised how monitoring can be applied or extended for increasing numbers of cases, variables and tuning parameters.

关键词： deterministic algorithm minimum covariance determinant nearby contamination S-estimator

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algorithm for calculating the roots of polynomials with coefficients in the ring of polynomials over an arbitrary integral domain

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MATHEMATICAL NOTES 2009年第1-2期85卷 68-81页

作者： Maevskii, A. E. So Fed Univ Rostov Na Donu Russia

A deterministic algorithm for calculating the roots of polynomials in one variable with coefficients in the ring of polynomials in several variables over an arbitrary integral domain is constructed. An estimate for th... 详细信息

关键词： root of a polynomial deterministic algorithm integral domain polynomial ring arithmetic complexity algebraic-geometric code principal prime ideal

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deterministically generating Picard groups of hyperelliptic curves over finite fields

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JOURNAL OF NUMBER THEORY 2016年 160卷 739-758页

作者： Kosters, Michiel UCI Irvine CA 92697 USA

Let is an element of > 0. In this article we will present a deterministic algorithm which does the following. The input is a hyperelliptic curve C of genus g over a finite field k of cardinality q given by y(2) + h(x)y = f (x) such that the x-coordinate map is ramified at infinity. In time 0(g(2+is an element of) q(1/2+is an element of)) the algorithm outputs a set of generators of the Picard group Pic(k)(0)(C). This extends results which others have obtained when g = 1. In this article we introduce a combinatorial tool, the shape parameter, which we use together with character sum estimates from class field theory to deduce the statement. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Picard group Hyperelliptic curve Finite field Shape parameter deterministic algorithm

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Sublinear Time Motif Discovery from Multiple Sequences

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algorithmS 2013年第4期6卷 636-677页

作者： Fu, Bin Fu, Yunhui Xue, Yuan Univ Texas Pan Amer Dept Comp Sci 1201 W Univ Dr Edinburg TX 78539 USA

In this paper, a natural probabilistic model for motif discovery has been used to experimentally test the quality of motif discovery programs. In this model, there are k background sequences, and each character in a background sequence is a random character from an alphabet, Sigma. A motif G = g(1)g(2) center dot center dot center dot g(m) is a string of m characters. In each background sequence is implanted a probabilistically-generated approximate copy of G. For a probabilistically-generated approximate copy b(1)b(2) center dot center dot center dot b(m) of G, every character, b(i), is probabilistically generated, such that the probability for b(i) not equal = g(i) is at most alpha. We develop two new randomized algorithms and one new deterministic algorithm. They make advancements in the following aspects: (1) The algorithms are much faster than those before. Our algorithms can even run in sublinear time. (2) They can handle any motif pattern. (3) The restriction for the alphabet size is a lower bound of four. This gives them potential applications in practical problems, since gene sequences have an alphabet size of four. (4) All algorithms have rigorous proofs about their performances. The methods developed in this paper have been used in the software implementation. We observed some encouraging results that show improved performance for motif detection compared with other software.

关键词： motif discovery sublinear time randomized algorithm deterministic algorithm

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Determination of eigenmodes by using Cauchy's integral formula

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ELECTRONICS LETTERS 1998年第19期34卷 1865-1866页

作者： Conradi, O Fern Univ Hagen D-58084 Hagen Germany

A deterministic method is proposed for the computation of zeros within a complex plane by using Cauchy's theorem. The integrals have been calculated for different numbers of integration points by the Gaussian integration formula. As an example, several eigenmodes of a TE/TM mode splitter have been determined by the proposed method.

关键词： Gaussian integration poles and zeros eigenvalues and eigenfunctions integration eigenmodes deterministic algorithms complex plane zeros deterministic algorithm TE/TM mode splitter Cauchy integral formula Linear algebra (numerical analysis) Numerical integration and differentiation

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A principle of stream arrangement based on uniformity factor for heat exchanger networks synthesis

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APPLIED THERMAL ENGINEERING 2013年第2期61卷 93-100页

作者： He, Qiaole Cui, Guomin Shanghai Univ Sci & Technol Sch Energy & Power Engn Shanghai 20093 Peoples R China

Heat exchanger network synthesis (HENS) has been one of the most extensively studied synthesis problem in chemical engineering. The methodologies commonly used to solve a heat exchanger network (HEN) problem are based on heuristic methods instead of deterministic methods which, theoretically, only with feasible initial HEN configuration enumeration, can find global optima. But enumeration always leads to unacceptable computational time. This paper presents a principle of process stream arrangement and the related deterministic algorithm for synthesizing HEN. The implementation of the principle based on the uniformity factor which proposed to describe the degree of the energy distribution of HEN configuration under different process stream arrangement is aimed to decrease the space of configuration enumeration. Thus, through the part of smaller uniformity factor network structure are optimized, deterministic methods are effectively suitable for medium-size problem with global optima in a reasonable brief computational time. Notably, using examples of 8SP1, 10SP, 10SP1, 14SP1, we demonstrate that our approach finds superior networks compared to those known in the literature. (C) 2013 Published by Elsevier Ltd.

关键词： Heat exchanger network synthesis Stream arrangement Uniformity factor deterministic algorithm

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