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

2025-11-24

AI Summary

Conventional permutation representations suffer from low representation efficiency in combinatorial optimization over permutation spaces (e.g., scheduling, routing), leading to excessive invalid search and slow convergence. Method: This paper systematically investigates the theoretical advantages and practical efficacy of Lehmer coding (i.e., inversion vectors) over standard permutation encodings. We propose a constraint-free Lehmer encoding scheme and, for the first time, establish rigorous theoretical connections between local perturbations in the Lehmer space and both inversion count and permutation distance metrics. Search efficiency gains are quantified via runtime analysis. Contribution/Results: Through theoretical analysis and empirical evaluation on sorting and quadratic assignment problems, we demonstrate that Lehmer coding significantly reduces invalid search, accelerates convergence, and consistently outperforms standard permutation encodings across multiple classical evolutionary algorithms. Our work provides an analytically grounded, empirically validated theoretical framework and an efficient implementation paradigm for permutation representation selection.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Analyzing Lehmer codes for efficient permutation space search
• Comparing inversion vector encodings with classical representations mathematically
• Evaluating practical performance on ordering and assignment problems

Innovation

• Using inversion vectors for permutation space representation
• Applying mathematical runtime analysis for efficiency comparison
• Linking inversion code changes to classical permutation measures