Online Disjoint Set Covers: Randomization is not Necessary

TL;DR

This paper introduces a deterministic online algorithm for the disjoint set covers problem that achieves an O(log^2 n) competitive ratio, matching randomized algorithms and extending derandomization techniques to new challenges.

Contribution

A novel potential function-based derandomization method for online set cover problems that improves competitive ratios and handles complex probabilistic challenges.

Findings

Deterministic algorithm matches randomized performance.

Achieves O(log^2 n) competitive ratio.

Extends derandomization to coupon collector and unbounded OPT.

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.