Exponential Upper Bounds for the Runtime of Randomized Search Heuristics

Doerr, Benjamin

doi:10.1007/978-3-030-58115-2_43

Cited by 6 publications

(2 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As part of a broader analysis of single-trajectory search heuristics, it was found that the Metropolis algorithm can optimize all weakly monotonic pseudo-Boolean functions in at most exponential time Doerr (2021).…”

Section: Previous Workmentioning

confidence: 99%

Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem

Doerr,

Rajabi,

Witt

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (2005). More precisely, denoting by n, m, w max , and w min the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T 0 ≥ w max and multiplicative cooling schedule with factor 1 − 1/ℓ, where ℓ = ω(mn ln(m)), with probability at least 1 − 1/m computes in time O(ℓ(ln ln(ℓ) + ln(T 0 /w min ))) a spanning tree with weight at most 1 + κ times the optimum weight, where 1 + κ = (1+o(1)) ln(ℓm) ln(ℓ)−ln(mn ln(m)) . Consequently, for any ε > 0, we can choose ℓ in such a way that a (1+ε)-approximation is found in time O((mn ln(n)) 1+1/ε+o(1) (ln ln n+ln(T 0 /w min ))) with probability at least 1 − 1/m. In the special case of so-called (1 + ε)separated weights, this algorithm computes an optimal solution (again in time O((mn ln(n)) 1+1/ε+o(1) (ln ln n + ln(T 0 /w min )))), which is a significant speed-up over Wegener's runtime guarantee of O(m 8+8/ε ).

show abstract

Section: Previous Workmentioning

confidence: 99%

Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem

Doerr,

Rajabi,

Witt

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For the first time T ′ that the modified algorithm finds a solution with fitness at least n − a we thus obtain While an exponential runtime on OneMax is not an exciting performance, for the sake of completeness we note that the runtime of the simple GA on OneMax is not worse than exponential. A runtime of exp(O(n)) can be shown with the methods of [Doe20b] (with some adaptations). The key observation is that, similar to property (A) in [Doe20b, Theorem 3], if at some time t the population contains an individual x with some fitness at least n/3, then in the next iteration this individual is chosen as parent at least once with at least constant probability and, conditional on this, with probability Ω( 1 n ) a particular better Hamming neighbor of x is generated from x.…”

Section: Fitness Proportionate Selectionmentioning

confidence: 99%

Lower Bounds for Non-Elitist Evolutionary Algorithms via Negative Multiplicative Drift

Doerr

2020

Preprint

Self Cite

View full text Add to dashboard Cite

A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems -general results which translate an expected progress away from the target into a high hitting time.We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) negative drift in populations method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a (1 − ω(n −1/2 )) factor below the threshold. For the special case of algorithms using standard bit mutation with a random mutation rate (called uniform mixing in the language of hyper-heuristics), we prove the result stated by Dang and Lehre (PPSN 2016) and extend it to mutation rates other than Θ(1/n), which includes the heavy-tailed mutation operator proposed by Doerr, Le, Makhmara, and Nguyen (GECCO 2017). We finally apply our method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple genetic algorithm on OneMax for arbitrary population size. * Extended version of a paper appearing in the proceedings of PPSN 2020 [Doe20c]. This version contains all proofs (for reasons of space, in [Doe20c] only Lemma 1 and Theorem 2 were proven), a new section on standard bit mutation with random mutation rates, and several additional details.

show abstract