在线集合多重覆盖问题的随机筛选逼近
Approximating the Online Set Multicover Problems Via Randomized Winnowing
摘要：
本文研究了加权在线集合k-多重覆盖问题。在这个问题中，我们有一个元素的宇宙V，一个V的子集族S，S中的每一个集合都有一个正的实际成本，还有一个“覆盖因子”（正整数）k。元素的子集以任意顺序在线呈现。当每个元素，比如i，被呈现时，我们还被告知所有（至少k个）集合的集合以及i所属的它们的成本，如果需要，我们需要从这些包含i的集合中选择额外的集合，使得我们所选择的集合的集合包含至少k个包含元素i的集合。我们的目标是最小化所选集合的总成本（我们的算法和竞争比界限可以扩展到一个集合最多可以被选择预定次数而不是一次的情况;为了简单起见，我们没有报告这些扩展，也因为它们与激励我们工作的生物学应用无关）。在本文中，我们描述了一个新的随机算法的基础上的随机版本的Littlestone的筛选方法的在线多覆盖问题。该算法推广和改进了一些较早的结果，改进了N。阿隆湾Awerbuch，Y. Azar，N. Buchbinder和J. Naor.我们还讨论了一般$k$的确定性算法的竞争比的下界。
英文摘要：
In this paper, we consider the weighted online set k-multicover problem. In this problem, we have a universe V of elements, a family S of subsets of V with a positive real cost for every set in S and a "coverage factor" (positive integer) k. A subset of elements are presented online in an arbitrary order. When each element, say i, is presented, we are also told the collection of all (at least k) sets and their costs to which i belongs and we need to select additional sets from these sets containing i, if necessary, such that our collection of selected sets contains at least k sets that contain the element i. The goal is to minimize the total cost of the selected sets (our algorithm and competitive ratio bounds can be extended to the case when a set can be selected at most a pre-specified number of times instead of just once; we do not report these extensions for simplicity and also because they have no relevance to the biological applications that motivated our work). In this paper, we describe a new randomized algorithm for the online multicover problem based on a randomized version of the winnowing approach of Littlestone. This algorithm generalizes and improves some earlier results by N. Alon, B. Awerbuch, Y. Azar, N. Buchbinder, and J. Naor. We also discuss lower bounds on competitive ratios for deterministic algorithms for general $k$.
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。
