An Improved Deterministic Algorithm for the Online Min-Sum Set Cover Problem

TL;DR

This paper introduces the first deterministic online algorithm for the Min-Sum Set Cover problem with a competitive ratio independent of the number of elements, improving previous bounds and achieving optimality relative to fixed permutations.

Contribution

It presents a novel deterministic algorithm with an $O(r^2)$ competitive ratio for the online Min-Sum Set Cover problem, surpassing prior bounds and matching asymptotic optimality.

Findings

Achieves $O(r^2)$ competitive ratio, independent of $n$

Surpasses previous bounds of $O(r^{3/2} imes \sqrt{n})$ and $O(r^4)$

Attains asymptotically optimal $O(r)$ ratio compared to fixed permutations

Abstract

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.