Random-Order Interval Selection (2407.20941v1)

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 show how to utilize random-order arrivals to extract a random bit with a worst case bias of 2/3, when there are at least two distinct item types. We use this bit to derandomize the barely random algorithm of [Fung et al. 2014] and get a deterministic 3-competitive algorithm for single-length interval selection with arbitrary weights.

• The paper shows that a simple greedy algorithm achieves a 2.5-competitive ratio in the random-order model for online interval selection.
• It introduces a revocable decision framework and a detailed charging scheme to optimize interval selection beyond adversarial models.
• The research uncovers a method for extracting random bits with bounded bias, paving the way for algorithm derandomization in similar settings.

Analysis of the Random-Order Interval Selection Problem

The paper addresses the complex problem of online unweighted interval selection, focusing on maximizing the number of non-conflicting intervals that an algorithm can accept. This research extends existing frameworks by examining the problem in the random-order model, which diverges from traditional adversarial settings by considering instances where the order of interval arrivals is randomized.

Competitive Ratio in the Random-Order Model

A substantial contribution of the research is the analysis of the competitive ratio in the random-order model for interval selection. Historically, competitive analysis in adversarial scenarios presents challenges, notably the well-documented Ω(n) lower bound on the competitive ratio, even when employing randomized algorithms. The current paper demonstrates that in a random-order setting, where the adversary selects the instance but the sequence of arrivals is random, an optimal competitive ratio is significantly reduced. The authors show that a simple greedy algorithm, which is 2k-competitive in the adversarial model, achieves a 2.5-competitive ratio when intervals arrive randomly. This reduction in competitive ratio highlights the advantages of utilizing random-order models for certain problem settings.

Methodology and Analytical Framework

The authors explore a variety of models including real-time, any-order, and random-order arrivals. The analytical framework pivots on revocable decisions—a scenario where interval revocation is permissible to optimize ultimate outcomes. This contrasts with traditional online decision-making, which assumes irrevocable decisions. In the context of random-order arrivals, this flexibility allows algorithms to refine their solutions dynamically, leading to potentially better performance metrics, particularly in maximizing the number of non-conflicting intervals.

The computational strategies delineated are based on an intricate charging scheme that maps optimal intervals to those accepted by the algorithm. This mapping is recursively defined through the execution of the algorithm and helps in analyzing and bounding the competitive ratio. For intervals with two or more different lengths, the research presents a detailed methodology for determining upper bounds on the expected transfer charge, which forms a core component of the competitive analysis.

Random Bit Extraction from Randomized Arrivals

The paper ventures into the theoretical implications of leveraging random arrivals to extract random bits with a bounded bias. Specifically, it describes a process to extract a single random bit from the arrival order with a worst-case bias of 2/3, contingent upon the presence of at least two distinct item types. This technique is then applied to derandomize the algorithm in cases where interval selection problems involve single-length intervals with arbitrary weights.

Implications and Future Directions

The implications of this research are notable in both theoretical and practical domains. The reduced competitive ratio in the random-order setting suggests that certain online problems can be managed more effectively when inputs arrive in a non-adversarial, stochastic manner. This insight extends beyond the field of interval scheduling to other combinatorial selection problems potentially benefiting from similar treatment.

Additionally, the exploration into random bit extraction paves the way for more profound inquiries into harnessing randomness from inherently random processes, thereby potentially simplifying algorithm designs that currently rely on genuine random bits.

In future studies, an intriguing direction is the investigation of whether better than 2-competitive deterministic algorithms for interval selection problems can exist under random-order conditions. Furthermore, exploring the effectiveness of randomized algorithms in this model poses an open question to ascertain if additional benefits can be achieved when moving beyond determinism.

In summary, the paper significantly advances the understanding of interval selection in the random-order model, offering robust analytical methods and broadening the horizons for how randomness in problem settings can improve algorithmic performance. This work exemplifies the potential to unlock new efficiencies and insights across various domains where online decision-making and optimization are critical.