The integrality number of an integer program
The integrality number of an integer program
复制标题
整数程序的完整性数
DOI：
10.1007/s10107-021-01651-0
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发表时间：
2021
影响因子：
2.7
通讯作者：
R. Weismantel
中科院分区：
文献类型：
--
作者：
Joseph Paat;Miriam Schlöter;R. Weismantel
引入了整数规划(IP)的整数概念。粗略地说，整数是通过混合整数(MIP)松弛来求解IP所需的最小整数约束数。这个数的一个显著性质是它在约束矩阵的么模变换下的不变性。考虑到约束矩阵的最大次要Δ\DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{waysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setLength{\oddsidemargin}{-69pt}\Begin{Document}$\varDelta$$\end{Document}，我们的分析允许我们做出如下形式的陈述：存在一个数字τ(Δ)\DocumentClass[12pt]{Minimum}\Usepackage{amsath}\usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setLong{\oddsidemarin}{-69pt}\Begin{Document}$$\tau(\varDelta)$\End{Document}使得具有n个多个变量和n+n/τ(Δ的IP\DocumentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$n+\Sqrt{n/\tau(\varDelta)}$$\end{Document}许多不等式约束可以通过少于n个整数约束的MIP松弛来解决。从我们的结果可以得出，仅由n个约束定义的IP可以通过具有O(Δ)\DocumentCLASS[12pt]{Minimum}\Usepackage{amsath}\Usepackage{amsFonts}\Usepackage{amssymb}\Usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\set long{\oddsidemargin}{-69pt}\Begin{Document}$O(\rSqt{\varDelta})$$\end{Document}多个整数约束的MIP松弛来求解。
We introduce the integrality number of an integer program (IP). Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta $$\end{document} of the constraint matrix, our analysis allows us to make statements of the following form: there exists a number τ(Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (\varDelta )$$\end{document} such that an IP with n many variables and n+n/τ(Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n + \sqrt{n /\tau (\varDelta )}$$\end{document} many inequality constraints can be solved via a MIP relaxation with fewer than n integer constraints. From our results it follows that IPs defined by only n constraints can be solved via a MIP relaxation with O(Δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\sqrt{\varDelta })$$\end{document} many integer constraints.