Fitness Probability Distribution of Bit-Flip Mutation

TL;DR

This paper derives exact probability distributions of fitness values after bit-flip mutation in binary strings, using polynomial expressions, with implications for analyzing the runtime of evolutionary algorithms.

Contribution

It introduces a novel method to compute the exact fitness distribution using Krawtchouk polynomials, applicable to both simple and complex optimization problems.

Findings

Exact polynomial expressions for fitness distributions are derived.

Closed-form solutions provided for Onemax and MAX-SAT problems.

Implications discussed for runtime analysis of evolutionary algorithms.

Abstract

Bit-flip mutation is a common mutation operator for evolutionary algorithms applied to optimize functions over binary strings. In this paper, we develop results from the theory of landscapes and Krawtchouk polynomials to exactly compute the probability distribution of fitness values of a binary string undergoing uniform bit-flip mutation. We prove that this probability distribution can be expressed as a polynomial in p, the probability of flipping each bit. We analyze these polynomials and provide closed-form expressions for an easy linear problem (Onemax), and an NP-hard problem, MAX-SAT. We also discuss some implications of the results for runtime analysis.