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 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

How to Use the Metropolis Algorithm for Multi-Objective Optimization?

“…Other algorithms were conducted with 50 independent runs and terminated when the full Pareto front was covered. We set α = 3 as in the paper (Wang, Zheng, and Doerr 2024) standard bit-wise mutation for NSGA-II, and set µ = n − 1 (at least n − 1 as suggested in Theorem 18).…”

“…Oliveto et al (2018) proved that the Metropolis algorithm solves their VALLEY function (which is different from the VALLEY problem defined in (Droste, Jansen, and Wegener 2000)) more efficiently than simple evolutionary algorithms (EAs). Wang, Zheng, and Doerr (2024) showed that the Metropolis algorithm solves the DECEPTIVELEADINGBLOCKS (DLB) function in expected time of O(n 2 ), while all (1+1)-elitist unary unbiased black-box algorithms have an expected runtime of Ω(n 3 ).…”

“…In this paper, we take several steps towards theoretically analyzing multiobjective Metropolis algorithms. Since an advantage of the Metropolis algorithm was proven for the single-objective DLB function (Wang, Zheng, and Doerr 2024), we construct a bi-objective counterpart of the DLB problem following a general construction method from (Laumanns, Thiele, and Zitzler 2004). We call our new benchmark DLTB, for deceptive leading blocks for ones and deceptive trailing blocks for zeros.…”