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 $\Omega(r)$ is a lower bound on the performance of any deterministic online algorithm.

翻译：我们研究在线偏好聚合的基本模型，其中算法维护一个包含$n$个元素的有序列表。输入为偏好集合$R_1, R_2, \dots, R_t, \dots$的数据流。当观察到$R_t$且对未来集合无任何先验知识时，算法需对元素重新排序（改变列表顺序），使得$R_t$中至少有一个元素位于列表前端附近。产生的代价包括列表更新代价（相邻列表元素的交换次数）和访问代价（$R_t$中首个元素在列表中的位置）。该场景自然出现在如利用网店顾客聚合偏好对商品进行排序等应用中。该问题的理论基础称为最小和集合覆盖。与先前工作（Fotakis等人，ICALP 2020，NIPS 2020）主要研究在线算法ALG相对于静态最优解（单一最优列表顺序）的性能不同，本文研究一个更具挑战性的变体，其中基准是已证明更强的动态最优解OPT（可能也会修改列表顺序）。对于网店而言，这意味着用户群体的聚合偏好随时间演化。我们构造了一个计算高效的随机算法，其竞争比（ALG与OPT的代价比）为$O(r^2)$，并证明了确定性$O(r^4)$-竞争算法的存在性。此处$r$是集合$R_t$的最大基数。这是首个竞争比不依赖于$n$的算法：此前该问题的最优算法为$O(r^{3/2} \cdot \sqrt{n})$-竞争，且$\Omega(r)$是任何确定性在线算法性能的下界。