Random-Order Interval Selection

TL;DR

This paper analyzes the online unweighted interval selection problem in the random-order model, providing an improved competitive ratio bound and exploring the use of random arrivals for derandomization and extracting randomness.

Contribution

It extends the analysis of interval selection algorithms to the random-order model and introduces a method for extracting random bits from random arrivals to derandomize algorithms.

Findings

Upper bound of 2.5 on the competitive ratio in the random-order model.

A process for extracting biased random bits with bias 0.585 from random arrivals.

Application of the extraction process to derandomize algorithms for various problems.

Abstract

In the problem of online unweighted interval selection, the objective is to maximize the number of non-conflicting intervals accepted by the algorithm. In the conventional online model of irrevocable decisions, there is an Omega(n) lower bound on the competitive ratio, even for randomized algorithms [Bachmann et al. 2013]. In a line of work that allows for revocable acceptances, [Faigle and Nawijn 1995] gave a greedy 1-competitive (i.e. optimal) algorithm in the real-time model, where intervals arrive in order of non-decreasing starting times. The natural extension of their algorithm in the adversarial (any-order) model is 2k-competitive [Borodin and Karavasilis 2023], when there are at most k different interval lengths, and that is optimal for all deterministic, and memoryless randomized algorithms. We study this problem in the random-order model, where the adversary chooses the instance, but the online sequence is a uniformly random permutation of the items. We consider the same algorithm that is optimal in the cases of the real-time and any-order models, and give an upper bound of 2.5 on the competitive ratio under random-order arrivals. We also initiate a study of utilizing random-order arrivals to extract random bits with the goal of derandomizing algorithms. Besides producing simple algorithms, simulating random bits through random arrivals enhances our understanding of the comparative strength of the two models. Our main extraction process returns a bit with a worst-case bias of 2 - sqrt(2) = 0.585 and operates under the mild assumption that there exist at least two distinct items in the input. We motivate the applicability of this process by using it to simulate a number of barely random algorithms for interval selection (single-length arbitrary weights, and C-benevolent instances), the general knapsack problem, string guessing, minimum makespan, and job throughput scheduling.