On the Runtime of Randomized Local Search and Simple Evolutionary Algorithms for Dynamic Makespan Scheduling (1504.06363v1)

Abstract: Evolutionary algorithms have been frequently used for dynamic optimization problems. With this paper, we contribute to the theoretical understanding of this research area. We present the first computational complexity analysis of evolutionary algorithms for a dynamic variant of a classical combinatorial optimization problem, namely makespan scheduling. We study the model of a strong adversary which is allowed to change one job at regular intervals. Furthermore, we investigate the setting of random changes. Our results show that randomized local search and a simple evolutionary algorithm are very effective in dynamically tracking changes made to the problem instance.

• The paper offers a rigorous performance analysis of RLS and (1+1) EA in dynamic makespan scheduling with varying job processing times.
• It shows that under adversarial conditions, RLS achieves a low discrepancy in O(n min{log n, log R}) time while (1+1) EA requires O(n^(3/2)) time.
• For random job time fluctuations, the (1+1) EA attains low discrepancy in O(n^4 log n) time, underscoring the algorithms’ robust adaptability.

Analysis of Randomized Local Search and Simple Evolutionary Algorithms for Dynamic Makespan Scheduling

The paper "On the Runtime of Randomized Local Search and Simple Evolutionary Algorithms for Dynamic Makespan Scheduling" authored by Frank Neumann and Carsten Witt, offers a theoretical analysis of the performance of evolutionary algorithms in the context of dynamic makespan scheduling. This work is pivotal as it navigates the complexities of a dynamic environment where job processing times are subject to changes, a common scenario in real-world applications.

Overview

The authors focus on the makespan scheduling problem, a quintessential combinatorial optimization task where the objective is to minimize the maximum load on a set of machines. The problem is analyzed in a dynamic setting, where job times can be altered, offering a realistic adaptation of the traditional makespan model. They examine two specific algorithms: Randomized Local Search (RLS) and the (1+1) Evolutionary Algorithm (EA), evaluating their efficiency in adapting to changes stipulated by two models—an adversarial model and a random model.

The Adversarial Model

In this model, changes to job processing times are governed by an adversary who may modify the times at regular intervals. The paper provides rigorous proofs establishing bounds on the performance of RLS and (1+1) EA in this setting. For RLS, the expected time to achieve a discrepancy of at most the upper bound $U$ is shown to be $O(n \min\{\log n, \log R\})$, where $R = U/L$ denotes the ratio of the upper and lower bounds of job times. Similarly, the (1+1) EA achieves this discrepancy in expected $O(n^{3/2})$ time. These results underline the resilience of these algorithms against frequent arbitrary changes, suggesting their applicability in highly dynamic environments.

The Random Model

The paper also investigates a random model where job times fluctuate due to a fair random process rather than adversarial intervention. The results are insightful, with (1+1) EA achieving a low discrepancy of $O(\log n)$ in $O(n^4 \log n)$ time. This indicates robust performance in situations where job times change in less predictable, stochastic patterns. The analysis further shows that when job changes occur continuously, both algorithms maintain a low discrepancy ratio relative to the makespan, underscoring their efficiency.

Implications and Future Directions

This analysis provides a foundational understanding of the behavior of simple evolutionary strategies in dynamic optimization, offering insights into their potential deployment in real-world scheduling tasks. The distinct performance expectations under both adversarial and random change models offer guidelines for selecting appropriate strategies based on the predictability of environmental changes. Future research could potentially explore extensions with more complex evolutionary operators or multi-objective scenarios, enhancing applicability and efficiency further.

Conclusion

Through an intricate computational complexity analysis, Neumann and Witt contribute substantially to the theoretical underpinnings of evolutionary algorithms in dynamic settings. Their work not only broadens the understanding of RLS and simple EA under dynamic conditions but also establishes a benchmark for future explorations into adaptive optimization methods. As the landscape of computational problems continues to evolve with more dynamic constraints, such analyses will be crucial for devising effective, adaptive algorithms.