We study the online variant of the Min-Sum Set Cover (MSSC) problem, a generalization of the well-known list update problem. In the MSSC problem, an algorithm has to maintain the time-varying permutation of the list of $n$ elements, and serve a sequence of requests $R_1, R_2, \dots, R_t, \dots$. Each $R_t$ is a subset of elements of cardinality at most $r$. For a requested set $R_t$, an online algorithm has to pay the cost equal to the position of the first element from $R_t$ on its list. Then, it may arbitrarily permute its list, paying the number of swapped adjacent element pairs. We present the first constructive deterministic algorithm for this problem, whose competitive ratio does not depend on $n$. Our algorithm is $O(r^2)$-competitive, which beats both the existential upper bound of $O(r^4)$ by Bienkowski and Mucha [AAAI '23] and the previous constructive bound of $O(r^{3/2} \cdot \sqrt{n})$ by Fotakis et al. [ICALP '20]. Furthermore, we show that our algorithm attains an asymptotically optimal competitive ratio of $O(r)$ when compared to the best fixed permutation of elements.

翻译：我们研究了在线最小和集合覆盖（MSSC）问题的变体，该问题是著名的列表更新问题的一种推广。在MSSC问题中，算法需要维护一个由$n$个元素组成的时变排列，并处理一系列请求$R_1, R_2, \dots, R_t, \dots$。每个$R_t$是元素的一个子集，其基数至多为$r$。对于请求集合$R_t$，在线算法需支付与其列表中$R_t$第一个元素位置相等的代价。然后，算法可以任意重排其列表，并支付交换相邻元素对的数量。我们提出了该问题的第一个构造性确定性算法，其竞争比不依赖于$n$。我们的算法是$O(r^2)$-竞争的，这优于Bienkowski和Mucha [AAAI '23] 的$O(r^4)$存在性上界以及Fotakis等人 [ICALP '20] 先前构造性$O(r^{3/2} \cdot \sqrt{n})$上界。此外，我们证明，在与元素最优固定排列比较时，我们的算法达到了渐近最优的$O(r)$竞争比。