Choosing the Right Algorithm With Hints From Complexity Theory

TL;DR

This paper uses complexity theory insights to guide the selection of optimization algorithms, demonstrating that the Metropolis algorithm and sig-cGA are highly effective for the DLB benchmark, outperforming traditional methods.

Contribution

It introduces a complexity-theoretic approach to algorithm selection, proving the effectiveness of the Metropolis algorithm and sig-cGA for the DLB problem, and establishes runtime bounds for various algorithms.

Findings

Metropolis algorithm solves DLB in quadratic time

sig-cGA solves DLB in O(n log n) time with high probability

Metropolis outperforms other algorithms on the DLB benchmark

Abstract

Choosing a suitable algorithm from the myriads of different search heuristics is difficult when faced with a novel optimization problem. In this work, we argue that the purely academic question of what could be the best possible algorithm in a certain broad class of black-box optimizers can give fruitful indications in which direction to search for good established optimization heuristics. We demonstrate this approach on the recently proposed DLB benchmark, for which the only known results are $O(n^3)$ runtimes for several classic evolutionary algorithms and an $O(n^2 \log n)$ runtime for an estimation-of-distribution algorithm. Our finding that the unary unbiased black-box complexity is only $O(n^2)$ suggests the Metropolis algorithm as an interesting candidate and we prove that it solves the DLB problem in quadratic time. Since we also prove that better runtimes cannot be obtained in the class of unary unbiased algorithms, we shift our attention to algorithms that use the information of more parents to generate new solutions. An artificial algorithm of this type having an $O(n \log n)$ runtime leads to the result that the significance-based compact genetic algorithm (sig-cGA) can solve the DLB problem also in time $O(n \log n)$ with high probability. Our experiments show a remarkably good performance of the Metropolis algorithm, clearly the best of all algorithms regarded for reasonable problem sizes.