Feasibility of integer knapsacks

Aliev, Iskander ORCID: https://orcid.org/0000-0002-2206-9207 and Henk, Martin 2010. Feasibility of integer knapsacks. Siam Journal of Optimization 20 (6) , pp. 2978-2993. 10.1137/090778043

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Abstract

Given a matrix $A\in\mathbb{Z}^{m\times n}$ satisfying certain regularity assumptions, we consider the set $\mathcal{F}(A)$ of all vectors $\boldsymbol{b}\in\mathbb{Z}^m$ such that the associated knapsack polytope $P(A,\boldsymbol{b})=\{\boldsymbol{x}\in\mathbb{R}^n_{\geq0}:A\boldsymbol{x}=\boldsymbol{b}\}$ contains an integer point. When $m=1$ the set $\mathcal{F}(A)$ is known to contain all consecutive integers greater than the Frobenius number associated with A. In this paper we introduce the diagonal Frobenius number $\mathrm{g}(A)$ which reflects in an analogous way feasibility properties of the problem and the structure of $\mathcal{F}(A)$ in the general case. We give an optimal upper bound for $\mathrm{g}(A)$ and also estimate the asymptotic growth of the diagonal Frobenius number on average. Read More: http://epubs.siam.org/doi/abs/10.1137/090778043

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Additional Information:	Pdf uploaded in accordance with publisher's policy at http://www.sherpa.ac.uk/romeo/issn/1052-6234/ (accessed 27/02/2014).
Publisher:	Society of Industrial and Applied Mathematics
ISSN:	1052-6234
Date of First Compliant Deposit:	30 March 2016
Last Modified:	14 May 2023 08:29
URI:	https://orca.cardiff.ac.uk/id/eprint/24988

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