Online Randomness Extraction: Simulating Barely Random Algorithms in the Random Order Arrival Model

TL;DR

This work investigates whether randomness in the arrival order can be harnessed to derandomize online algorithms in the random-order model (ROM). It introduces three 1-bit randomness-extraction processes and culminates in COMBINE, a method that achieves a worst-case bias of $2-\sqrt{2}$ under mild distinct-item assumptions. The extracted bits are used to de-randomize a suite of barely-random online algorithms across problems such as binary string guessing, knapsack (general and proportional weights), interval scheduling (including single-length and benevolent weight variants), and unweighted throughput, yielding new constant ROM competitive ratios in several cases. The results provide a constructive step toward understanding the relation between randomized online algorithms and deterministic ROM algorithms, and they outline clear avenues for extending randomness extraction to multiple bits and broader applications in ROM settings.

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 de-randomizing 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 sequence) 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 arbitrary weights, as well as monotone, C-benevolent and D-benevolent weighted instances), the proportional and general knapsack problems, binary string guessing, and unweighted job throughput scheduling. 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 towards understanding the relation between randomized online algorithms and deterministic ROM algorithms.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Lemma 3
• Theorem 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8: Lemma 2.2 chrobak2007online
• Theorem 9: Theorem 3.1 chrobak2007online
• Lemma 10