On scenario aggregation to approximate robust combinatorial optimization problems
On scenario aggregation to approximate robust combinatorial optimization problems
复制标题
关于近似鲁棒组合优化问题的场景聚合
DOI：
10.1007/s11590-017-1206-x
复制
发表时间：
2017
影响因子：
1.6
通讯作者：
M. Goerigk
中科院分区：
文献类型：
--
作者：
A. Chassein;M. Goerigk
由于大多数具有离散不确定集的鲁棒组合最小最大和最小-最大后悔问题都是NP难的，因此对逼近算法和可逼近界的研究一直是最近卓有成效的工作领域。一种简单而著名的近似算法是中点法，其中取所有情况的平均值，并解决名义类型的问题。尽管这种方法很简单，但它仍然在一系列问题上给出了众所周知的界，例如鲁棒最短路径或鲁棒指派问题。本文提出了一种基于情景集结的中点方法的简单扩展，它将当前的最佳K-近似结果改进为(εK)\DocumentClass[12pt]{Minimum}\Usepackage{amsath}\Usepackage{waysym}\usepackage{amsFonts}\Usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setLength{\oddsidemargin}{-69pt}\Begin{Document}$(\varepsilon K)$\end{Document}-近似；0\Docentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$\varepsilon>；0$$\end{Document}。该方法不仅适用于最小-最大后悔问题，也适用于最小-最大后悔问题。
As most robust combinatorial min–max and min–max regret problems with discrete uncertainty sets are NP-hard, research in approximation algorithm and approximability bounds has been a fruitful area of recent work. A simple and well-known approximation algorithm is the midpoint method, where one takes the average over all scenarios, and solves a problem of nominal type. Despite its simplicity, this method still gives the best-known bound on a wide range of problems, such as robust shortest path or robust assignment problems. In this paper, we present a simple extension of the midpoint method based on scenario aggregation, which improves the current best K-approximation result to an (εK)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\varepsilon K)$$\end{document}-approximation for any desired ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}. Our method can be applied to min–max as well as min–max regret problems.