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Lattice based extended formulations for integer linear equality systems

机译:基于格的整数线性等式系统的扩展公式

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We study different extended formulations for the set with in order to tackle the feasibility problem for the set Pursuing the work of Aardal, Lenstra et al. using the reformulation , our aim is to derive reformulations of the form with 0 ≤ s ≤ n − m where preferably all the coefficients of P are small compared to the coefficients of A and T. In such cases the new variables μ appear to be good branching directions, and in certain circumstances permit one to deduce rapidly that the instance is infeasible. We give a polynomial time algorithm for identifying such P, T if possible, and for the case that A has one row a we analyze the reformulation when s = 1, that is, one μ-variable is introduced. In particular, we determine the integer width of the extended formulations in the direction of the μ-variable, and derive a lower bound on the Frobenius number of a. We conclude with some preliminary tests to see if the reformulations are effective when the number s of additional constraints and variables is limited. Keywords Integer programming - Lattice basis reduction - Integer width - Frobenius number Mathematics Subject Classification (2000) 90C10 - 45A05 - 11Y50 This work was partly carried out within the framework of ADONET, a European network in Algorithmic Discrete Optimization, contract no. MRTN-CT-2003-504438. The first author is financed in part by the Dutch BSIK/BRICKS project. The research was carried out in part while the second author visited CWI, Amsterdam with the support of the NWO visitor grant number B 61-556.
机译:为了解决布景的可行性问题,我们研究了布景的不同扩展公式。追求Aardal,Lenstra等人的工作。使用重新制定公式,我们的目的是得出具有0≤s≤n-m形式的重新制定公式,其中与A和T的系数相比,最好P的所有系数都较小。在这种情况下,新变量μ看起来很好分支方向,并且在某些情况下允许人们迅速推断出该实例不可行。我们给出了多项式时间算法来识别这样的P,T(如果可能),并且对于A有一行a的情况,我们分析了s = 1时的重新公式化,即引入了一个μ变量。特别是,我们确定扩展公式在μ变量方向上的整数宽度,并得出a的Frobenius数的下界。我们以一些初步测试作为结束,以查看当其他约束和变量的数量受到限制时,重新制定是否有效。关键字整数编程-晶格基约化-整数宽度-Frobenius数数学主题分类(2000)90C10-45A05-11Y50这项工作部分在ADONET框架内进行,ADONET是欧洲算法离散优化网络,合同号为。 MRTN-CT-2003-504438。第一作者由荷兰BSIK / BRICKS项目提供部分资金。该研究的一部分是在第二位作者在NWO游客资助号B 61-556的支持下访问了阿姆斯特丹CWI的同时进行的。

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