Online Disjoint Set Covers: Randomization is not Necessary
Marcin Bienkowski, Jarosław Byrka, Łukasz Jeż
TL;DR
This work studies online disjoint set covers in hypergraphs, where incoming hyperedges must be colored on the fly to maximize the number of colors that form full set covers. It introduces Rand, a randomized baseline inspired by prior work, and develops a novel potential-based derandomization that operates directly on the input, handling challenges such as unbounded OPT and coupon-collector-type progress. The authors obtain a deterministic online algorithm Det with $O(\log^2 n)$-competitiveness, exponentially improving the prior $O(n)$ bound and matching the best known randomized performance. The technique extends online derandomization by combining a carefully constructed potential with an online analogue of conditional probabilities, offering a potential toolkit for derandomizing other online randomized algorithms. The results have implications for tasks like sensor activation and resource allocation in online settings, where guarantees on the number of fully utilized colors translate to robust cover decompositions.
Abstract
In the online disjoint set covers problem, the edges of a hypergraph are revealed online, and the goal is to partition them into a maximum number of disjoint set covers. That is, n nodes of a hypergraph are given at the beginning, and then a sequence of hyperedges (subsets of [n]) is presented to an algorithm. For each hyperedge, an online algorithm must assign a color (an integer). Once an input terminates, the gain of the algorithm is the number of colors that correspond to valid set covers (i.e., the union of hyperedges that have that color contains all n nodes). We present a deterministic online algorithm that is O(log^2 n)-competitive, exponentially improving on the previous bound of O(n) and matching the performance of the best randomized algorithm by Emek et al. [ESA 2019]. For color selection, our algorithm uses a novel potential function, which can be seen as an online counterpart of the derandomization method of conditional probabilities and pessimistic estimators. There are only a few cases where derandomization has been successfully used in the field of online algorithms. In contrast to previous approaches, our result extends to the following new challenges: (i) the potential function derandomizes not only the Chernoff bound, but also the coupon collector's problem, (ii) the value of OPT of the maximization problem is not bounded a priori, and (iii) we do not produce a fractional solution first, but work directly on the input.