arXiv 2209.048702022-09cs.DScs.LG

一种改进的在线最小和集覆盖算法

An Improved Algorithm For Online Min-Sum Set Cover

University of Wrocław（弗罗茨瓦夫大学）；University of Warsaw（华沙大学）

Marcin Bienkowski, Marcin Mucha

备注：: Presented at AAAI 2023

摘要：

我们研究了在线偏好聚合的基本模型，其中的算法维护一个有序的列表的$n$元素。输入是优选集合$R_1, R_2, \dots, R_t, \dots$的流。在看到$R_t$并且不知道任何未来的集合时，算法必须重新排列元素（改变列表排序），使得$R_t$的至少一个元素在列表前端附近被发现。所引起的成本是列表更新成本（相邻列表元素的交换次数）和访问成本（$R_t$的第一元素在列表上的位置）之和。这种场景自然会出现在应用程序中，例如使用商店客户的聚合偏好在在线商店中订购物品。这个问题的理论基础被称为最小和集覆盖。不像以前的工作（Fotakis等人，ICALP 2020，NIPS 2020）主要研究了在线算法ALG对静态最优解（单个最优列表排序）的性能，在本文中，我们研究了一个可以说更难的变体，其中基准是可证明更强的最优动态解OPT（也可以修改列表排序）。就在线商店而言，这意味着其用户群的聚合偏好会随着时间的推移而演变。我们构造了一个计算效率高的随机算法，其竞争比（ALG-OPT成本比）是$O(r^2)$，并证明了存在一个确定性的$O(r^4)$ -竞争算法。这里，$r$是集合$R_t$的最大基数。这是第一个算法的比例不依赖于$n$：以前最好的算法，这个问题是$O(r^{3/2} \cdot \sqrt{n})$ -竞争和$Ω(r)$是一个下界的任何确定性在线算法的性能。

英文摘要：

We study a fundamental model of online preference aggregation, where an algorithm maintains an ordered list of $n$ elements. An input is a stream of preferred sets $R_1, R_2, \dots, R_t, \dots$. Upon seeing $R_t$ and without knowledge of any future sets, an algorithm has to rerank elements (change the list ordering), so that at least one element of $R_t$ is found near the list front. The incurred cost is a sum of the list update costs (the number of swaps of neighboring list elements) and access costs (position of the first element of $R_t$ on the list). This scenario occurs naturally in applications such as ordering items in an online shop using aggregated preferences of shop customers. The theoretical underpinning of this problem is known as Min-Sum Set Cover. Unlike previous work (Fotakis et al., ICALP 2020, NIPS 2020) that mostly studied the performance of an online algorithm ALG against the static optimal solution (a single optimal list ordering), in this paper, we study an arguably harder variant where the benchmark is the provably stronger optimal dynamic solution OPT (that may also modify the list ordering). In terms of an online shop, this means that the aggregated preferences of its user base evolve with time. We construct a computationally efficient randomized algorithm whose competitive ratio (ALG-to-OPT cost ratio) is $O(r^2)$ and prove the existence of a deterministic $O(r^4)$-competitive algorithm. Here, $r$ is the maximum cardinality of sets $R_t$. This is the first algorithm whose ratio does not depend on $n$: the previously best algorithm for this problem was $O(r^{3/2} \cdot \sqrt{n})$-competitive and $Ω(r)$ is a lower bound on the performance of any deterministic online algorithm.

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。

摘要：

英文摘要：

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