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    188宝金博页面版: rounding of sequences and matrices, with applications

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    内容提示： Rounding of Sequences and Matrices,with ApplicationsBenjamin Doerr, Tobias Friedrich, Christian Klein, and Ralf OsbildMax-Planck-Institut f¨ ur Informatik, Saarbr¨ ucken, GermanyAbstract. We show that any real matrix can be rounded to an inte-ger matrix in such a way that the rounding errors of all row sums areless than one, and the rounding errors of all column sums as well as allsums of consecutive row entries are less than two. Such roundings can becomputed in linear time. This extends and improves pr...

    文档格式：PDF | 页数：14 | 浏览次数：14 | 上传日期：2021-04-03 16:49:48 | 文档星级：
    Rounding of Sequences and Matrices,with ApplicationsBenjamin Doerr, Tobias Friedrich, Christian Klein, and Ralf OsbildMax-Planck-Institut f¨ ur Informatik, Saarbr¨ ucken, GermanyAbstract. We show that any real matrix can be rounded to an inte-ger matrix in such a way that the rounding errors of all row sums areless than one, and the rounding errors of all column sums as well as allsums of consecutive row entries are less than two. Such roundings can becomputed in linear time. This extends and improves previous results onrounding sequences and matrices in several directions. It has particularapplications in just-in-time scheduling, where balanced schedules on ma-chines with negligible switch over costs are sought after. Here we extendexisting results to multiple machines and non-constant production rates.1IntroductionIn this paper, we analyze a rounding problem with connections to different areasin discrete mathematics, computer science, and operations research. Roughlyspeaking, we show that any real matrix can be rounded to an integer one in sucha way that the rounding errors of all row and column sums are less than one,and the rounding errors of all sums of consecutive row entries are less than two.Let m,n be positive integers. For some set S, we write Sm×nto denote theset of m × n matrices with entries in S. For real numbers a,b let [a..b] := {z ∈Z|a ≤ z ≤ b}.Theorem 1. Let X ∈ Rm×nhaving integral column sums. Then there is aY ∈ Zm×nsuch that∀j ∈ [1..n] :m?i=1(xij− yij) = 0,∀b ∈ [1..n],i ∈ [1..m] :???b?j=1(xij− yij)??? < 1.Such a matrix Y can be computed in time O(mn).It is easy to see that the second condition implies that for all a,b ∈ [1..n] andi ∈ [1..m] we have |?bextended to matrices having arbitrary column sums. See Section 3 for the details.Theorem 1 extends and improves a number of results from different applica-tions.j=a(xij− yij)| < 2. Also, the theorem can easily be

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