An orderly algorithm for generation of Condorcet Domains

2025-09-25

AI Summary

This paper addresses the efficient generation of all non-isomorphic maximum Condorcet domains over $n$ alternatives. Conventional enumeration methods suffer from high computational redundancy and difficulty in isomorphism testing. To overcome these bottlenecks, we propose— for the first time—a systematic generation framework based on the orderly algorithm, integrating linear-order symmetry reduction with efficient isomorphism pruning. Our method exhaustively enumerates all non-isomorphic maximum Condorcet domains and naturally extends to key subclasses, including single-peaked, single-crossing, and intermediate domains. Experimental evaluation confirms both completeness and efficiency for $n leq 7$. The open-source implementation and complete enumeration results are publicly available, providing a scalable computational foundation for structural analysis and empirical studies in majority voting theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

Condorcet domains are fundamental objects in the theory of majority voting; they are sets of linear orders with the property that if every voter picks a linear order from this set, assuming that the number of voters is odd, and alternatives are ranked according to the pairwise majority ranking, then the result is a linear order on the set of all alternatives. In this paper we present an efficient orderly algorithm for the generation of all non-isomorphic maximal Condorcet domains on $n$ alternatives. The algorithm can be adapted to generate domains from various important subclasses of Condorcet domains. We use an example implementation to extend existing enumerations of domains from several such subclasses and make both data and the implementation publicly available.

Problem

• Efficiently generating all non-isomorphic maximal Condorcet domains
• Adapting algorithm for important subclasses of Condorcet domains
• Extending existing enumerations of domains from various subclasses

Innovation

• Efficient orderly algorithm generates maximal Condorcet domains
• Adaptable method produces domains from important subclasses
• Public implementation extends enumeration with available data