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Beck, Matthias

doi:10.1023/a:1009853104418

Cited by 44 publications

(5 citation statements)

References 8 publications

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“…Our algorithm and the numerical study for it are based on the idea of computing a certain coefficient in the weighted geometric series via the Cauchy formula for complex path integrals. This connection was observed by Brion and Vergne (1997b), Beck (2000), Lasserre and Zeron (2002) and used for computations in, e.g., Beck and Pixton (2003). In addition, the methods have been applied successfully to special applications, such as networks flows (Baldoni-Silva et al 2004) and computational number theory (Baldoni et al 2014).…”

Section: Related Workmentioning

confidence: 83%

“…, z m . Under mild assumptions on A, we can apply multivariate techniques to give an explicit generating function for the solutions to Ax = b, x ∈ Z n + in the form of a power series in m variables, see Beck (2000), Lasserre andZeron (2002), andFriedrich (2016).…”

Section: Description Of the Methodsmentioning

confidence: 99%

“…While we can use (3) to count the number of solutions to a knapsack instance, thus connecting our work immediately to Baldoni-Silva et al (2004), Beck (2000), Lasserre and Zeron (2007), and others, the reformulation in (4) focuses on the feasibility setting. In particular, it is not necessary to compute the exact value for the integral in (4) to decide whether the problem is feasible or not.…”

Section: Path Integralsmentioning

confidence: 99%

“…As this section demonstrates, understanding the locations, orders and symmetries of the poles of (3) requires some deeper algebraic reasoning and structured research in this direction has been undertaken by Beck (2000Beck ( , 2004, Lasserre and Zeron (2007), and in Hirai et al (2020). We look to avoid such reasoning in our approach and instead address the problem from a numerical integration perspective.…”

Section: Dominant Polesmentioning

confidence: 99%

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Computing Optimized Path Integrals for Knapsack Feasibility

Daues

Friedrich

2022

INFORMS Journal on Computing

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A generating function technique for solving integer programs via the evaluation of complex path integrals is discussed from a theoretical and computational perspective. Applying the method to knapsack feasibility problems, it is demonstrated how the presented numerical integration algorithm benefits from a preoptimized path of integration. After discussing the algorithmic setup in detail, a numerical study is implemented to evaluate the computational performance of the preoptimized integration method, and the algorithmic parameters are tuned to a set of knapsack instances. The goal is to highlight the method’s computational advantage for hard knapsack instances. Summary of Contribution: A method for evaluating the feasibility of knapsack problems is discussed and connected to the existing theory on generating function techniques for computational integer optimization. Specifically, the number of solutions to knapsack instances is computed using numerical quadrature of complex path integrals. The choice of the path of integration is identified as an important degree of freedom, and it is shown that preoptimizing the path improves the computational performance for hard instances significantly. The scope of this work is to give a self-contained presentation of the mathematical theory, introduce and discuss path optimization as a presolving routine, and demonstrate the computational performance of the overall approach with the help of a detailed numerical study. The main goal is to highlight the computational advantage of the new algorithm for hard knapsack instances.

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Section: Related Workmentioning

confidence: 83%

Section: Description Of the Methodsmentioning

confidence: 99%

Section: Path Integralsmentioning

confidence: 99%

Section: Dominant Polesmentioning

confidence: 99%

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Computing Optimized Path Integrals for Knapsack Feasibility

Daues

Friedrich

2022

INFORMS Journal on Computing

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“…The residue theorem, which is also known as the Cauchy residue theorem [4][5][6], can help us calculate some definite integrals efficiently. Cauchy considered the difference of integrals along two paths with standard endpoints sandwiching a function pole in the middle, thus forming the residue concept [7].…”

Section: Introductionmentioning

confidence: 99%

On the Calculation of Several Definite Integrals by Residue Theorem

Huang¹,

Jiang²,

Yang³

et al. 2023

HSET

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The Cauchy’s residue theorem is one of the most important theorems in complex analysis at all times, and it is demonstrated that using the residue theorem is an easier and faster method to calculate some types of real improper integrals when the targeted integrals are hard or even impossible to deal with by conventional approaches. This paper considers several types of integrals including trigonometric integrals and integrals involving logarithmic function and power function. The general methods for calculating these integrals are presented and typical examples to illustrate how to use the methods are shown. The types of integrals in the paper are useful in many fields and have applications in the engineering and scientific research. In complex analysis, the residue theorem is a powerful tool for calculating the path integrals of analytic functions along closed curves, and can also be used to calculate the integrals of real functions.

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The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums

Beck

Diaz

Robins

2002

Journal of Number Theory

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We study the number of lattice points in integer dilates of the rational polytope P ¼ ðx 1 ; . . . ; x n Þ 2 R n 50 :where a 1 ; . . . ; a n are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a 1 ; . . . ; a n ; find the largest value of t (the Frobenius number) such that m 1 a 1 þ Á Á Á þ m n a n ¼ t has no solution in positive integers m 1 ; . . . ; m n : This is equivalent to the problem of finding the largest dilate tP such that the facet f P n k¼1 x k a k ¼ tg contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials Lð % P P; tÞ :¼ #ðtP \ Z n Þ and LðP8; tÞ :¼ #ðtP8 \ Z n Þ: Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier-Dedekind sums and use them to give new bounds for the Frobenius number. # 2002 Elsevier Science (USA)

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Untitled

Cited by 44 publications

References 8 publications

Computing Optimized Path Integrals for Knapsack Feasibility

Computing Optimized Path Integrals for Knapsack Feasibility

On the Calculation of Several Definite Integrals by Residue Theorem

The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums

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