Choosing the right algorithm with hints from complexity theory

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 heuristics. We demonstrate this approach on the recently proposed DLB benchmark. Our finding that the unary unbiased black-box complexity is only $O\left(n^2\right)$ suggests the Metropolis algorithm as an interesting candidate and we prove that it solves the DLB problem in quadratic time. We also prove that better runtimes cannot be obtained in the class of unary unbiased algorithms. We therefore shift our attention to algorithms that use the information of more parents to generate new solutions and find that the significance-based compact genetic algorithm can solve the DLB problem in time $O(n\log n)$.