Theoretical and Empirical Analysis of Lehmer Codes to Search Permutation Spaces with Evolutionary Algorithms

TL;DR

This paper analyzes the use of Lehmer codes (inversion vectors) for representing permutations in evolutionary algorithms, providing theoretical insights and empirical evidence of their efficiency in solving permutation-based problems.

Contribution

It offers a rigorous theoretical comparison between Lehmer codes and classical permutation representations, along with practical experiments demonstrating their effectiveness.

Findings

Lehmer codes facilitate constraint-free permutation representation.

Theoretical analysis shows efficiency advantages over classical encoding.

Experimental results confirm practical benefits in scheduling and assignment problems.

Abstract

A suitable choice of the representation of candidate solutions is crucial for the efficiency of evolutionary algorithms and related metaheuristics. We focus on problems in permutation spaces, which are at the core of numerous practical applications of such algorithms, e.g. in scheduling and transportation. Inversion vectors (also called Lehmer codes) are an alternative representation of the permutation space $S_n$ compared to the classical encoding as a vector of $n$ unique entries. In particular, they do not require any constraint handling. Using rigorous mathematical runtime analyses, we compare the efficiency of inversion vector encodings to the classical representation and give theory-guided advice on their choice. Moreover, we link the effect of local changes in the inversion code space to classical measures on permutations like the number of inversions. Finally, through experimental studies on linear ordering and quadratic assignment problems, we demonstrate the practical efficiency of inversion vector encodings.