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The Runtime of Random Local Search on the Generalized Needle Problem

Benjamin Doerr, Andrew James Kelley

TL;DR

An exact description of the expected runtime is derived, which also significantly improves the upper bound given by Doerr and Krejca, and asymptotic estimates of the expected runtime are described.

Abstract

In their recent work, C. Doerr and Krejca (Transactions on Evolutionary Computation, 2023) proved upper bounds on the expected runtime of the randomized local search heuristic on generalized Needle functions. Based on these upper bounds, they deduce in a not fully rigorous manner a drastic influence of the needle radius $k$ on the runtime. In this short article, we add the missing lower bound necessary to determine the influence of parameter $k$ on the runtime. To this aim, we derive an exact description of the expected runtime, which also significantly improves the upper bound given by C. Doerr and Krejca. We also describe asymptotic estimates of the expected runtime.

The Runtime of Random Local Search on the Generalized Needle Problem

TL;DR

An exact description of the expected runtime is derived, which also significantly improves the upper bound given by Doerr and Krejca, and asymptotic estimates of the expected runtime are described.

Abstract

In their recent work, C. Doerr and Krejca (Transactions on Evolutionary Computation, 2023) proved upper bounds on the expected runtime of the randomized local search heuristic on generalized Needle functions. Based on these upper bounds, they deduce in a not fully rigorous manner a drastic influence of the needle radius on the runtime. In this short article, we add the missing lower bound necessary to determine the influence of parameter on the runtime. To this aim, we derive an exact description of the expected runtime, which also significantly improves the upper bound given by C. Doerr and Krejca. We also describe asymptotic estimates of the expected runtime.
Paper Structure (9 sections, 15 theorems, 53 equations)

This paper contains 9 sections, 15 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Generalized Needle Functions
  3. Exact Runtimes
  4. Estimates for the Runtime
  5. Sub-linear Values of $k$
  6. Linear Values for $k$, Linearly Bounded Away From $n/2$
  7. $k$ Close to $n/2$
  8. $k$ Larger than $n/2$
  9. Conclusion

Key Result

Theorem 1

Let $n \in \mathbb{N}$ and $k \in [0..n]$. Let $i \in [0..n]$. Let $T(i)$ be the runtime of RLS on $\textsc{Needle}\xspace_{n,k}$ when starting with an initial solution having exactly $i$ ones. Then for $i \le n-k-1$ and $\mathop{\mathrm{E}}\nolimits[T(i)]=0$ for $i \in [n-k..n]$. Let $T$ be the runtime when starting with a random solution. Then

Theorems & Definitions (29)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['thm:exact_expectation']}
  • Theorem 5
  • Lemma 6
  • proof
  • ...and 19 more