An Improved Algorithm for Online Min-Sum Set Cover

Abstract

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, ..., R_t, ... 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 cost (the 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 that mostly studied the performance of an online algorithm ALG in comparison to 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) * n^(1/2))-competitive and Ω(r) is a lower bound on the performance of any deterministic online algorithm.