鲁棒最大最小优化问题的带约束覆盖算法
Thresholded Covering Algorithms for Robust and Max-Min Optimization
摘要：
鲁棒优化的一般问题是：明天可能出现的几种情况之一，但明天的东西比今天更昂贵。你今天应该提前购买什么，以便最坏情况下的成本（两天之和）最小化？Feige等人和Khandekar等人考虑了k-鲁棒模型，其中明天的可能结果由所有大小为k的需求子集给出，并分别给出了该模型中的集合覆盖问题、Steiner树和设施选址问题的算法。本文给出了k-鲁棒问题的一个简单直观的模板：“在建立了某个预期解之后，如果存在一个需求的增广成本大于某个阈值，则增广该预期解以覆盖该需求，然后重复”。在本文中，我们表明，该模板给我们改进的近似算法的k-鲁棒Steiner树和集覆盖，以及第一个近似算法的k-鲁棒Steiner森林，最小割和多割。我们所有的近似比（除了multicut）几乎都是最好的。作为我们技术的副产品，我们还得到了如下形式的最大-最小问题的算法：“给定一个覆盖问题实例，覆盖哪k个元素的成本最高？".
英文摘要：
The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
