Deterministically Simulating Barely Random Algorithms in the Random-Order Arrival Model

TL;DR

This paper develops a method to simulate barely random algorithms in the random-order model, providing insights into the power comparison between randomized and deterministic online algorithms across various problems.

Contribution

It introduces a deterministic process for extracting near-random bits from random arrivals, enabling the simulation of randomized algorithms in the ROM.

Findings

Achieves a bias of approximately 0.585 in randomness extraction.

Uses the process to simulate algorithms for weighted interval selection, knapsack, scheduling, and makespan minimization.

Provides a framework for understanding the comparative strength of randomized and deterministic algorithms in the ROM.

Abstract

Interest in the random-order model (ROM) leads us to 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 randomized online algorithms (with adversarial input sequences) and deterministic algorithms in the ROM. We consider three $1$-bit randomness extraction processes. Our best extraction process returns a bit with a worst-case bias of $2 - \sqrt{2} \approx 0.585$ 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 weighted interval selection (single-length with arbitrary weights, as well as monotone, C-benevolent and D-benevolent weighted instances), the proportional and general knapsack problems, job throughput scheduling, and makespan minimization. It is well known that there are many applications where a deterministic ROM algorithm significantly outperforms any randomized online algorithm (in terms of competitive ratios). The classic example is that of the secretary problem. We ask the following fundamental question: Is there any application for which a randomized algorithm outperforms any deterministic ROM algorithm? Motivated by this question, we view our randomness extraction applications as a constructive approach toward understanding the relationship between randomized online algorithms and deterministic algorithms in the ROM.