Metrics on Permutation Families Defined by a Restriction Graph

2025-07-08

AI Summary

This paper investigates the diameter problem for families of permutations defined by constraint graphs (directed acyclic graphs) under the ℓ∞ and Kendall–Tau metrics. The central result establishes a necessary and sufficient condition: the Kendall–Tau diameter achieves its combinatorial upper bound if and only if the partial order induced by the constraint graph has dimension at most two—thereby forging the first exact link between metric extremality and poset dimension. Building on this characterization, we devise an explicit algorithm to construct maximally distant permutation pairs. As corollaries, we obtain closed-form expressions for the metric diameters of classical structures including descent sets and Hessenberg varieties. Our approach integrates tools from poset theory, permutation combinatorics, and graph theory, enabling efficient computation. The results provide new theoretical foundations and algorithmic tools for combinatorial coding, fault-tolerant ranking, and metric embedding.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Understanding the metric structure of permutation families is fundamental to combinatorics and has applications in social choice theory, bioinformatics, and coding theory. We study permutation families defined by restriction graphs--oriented graphs that constrain the relative order of elements in valid permutations. For any restriction graph $G$, we determine the maximum distance achievable by two permutations under the $ell_infty$-metric and provide an explicit algorithm that constructs optimal permutation pairs. Our main contribution characterizes when the Kendall-Tau metric achieves its combinatorial upper bound: this occurs if and only if the poset induced by $G$ has dimension at most 2. When this condition holds, the extremal permutations form a minimal realizer of the poset, revealing a deep connection between metric geometry and poset dimension theory. We apply these results to classical permutation statistics including descent sets and Hessenberg varieties, obtaining explicit formulas and efficient algorithms for computing metric diameters.

Problem

• Study maximum distance in permutation families under l∞-metric
• Characterize when Kendall-Tau metric meets its upper bound
• Apply results to permutation statistics and metric diameters

Innovation

• Uses restriction graphs to define permutation families
• Determines maximum distance under $ell_infty$-metric
• Links Kendall-Tau metric to poset dimension theory