Diversity of Structured Domains via k-Kemeny Scores

2025-09-19

AI Summary

This paper investigates the *k*-Kemeny problem—minimizing the total number of adjacent swaps required to transform a given set of ordinal votes into at most *k* distinct total orders—within structured preference domains (e.g., single-peaked, single-crossing, group-separable, Euclidean). Through computational complexity analysis and combinatorial optimization, we establish that the problem remains NP-hard in nearly all such domains, even for *k* = 2. Building on the *k*-Kemeny score, we propose a novel diversity quantification framework that systematically measures and ranks the expressive capacity of different domains. Our results reveal that, despite structural constraints, these domains can support highly heterogeneous ranking distributions. The core contribution is a formal model linking structural preference diversity to computational intractability, accompanied by a principled, comparable, and quantifiable benchmark for evaluating domain expressiveness and algorithmic hardness.

Technology Category

• Knowledge Representation and Reasoning: Preferences
• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

In the k-Kemeny problem, we are given an ordinal election, i.e., a collection of votes ranking the candidates from best to worst, and we seek the smallest number of swaps of adjacent candidates that ensure that the election has at most k different rankings. We study this problem for a number of structured domains, including the single-peaked, single-crossing, group-separable, and Euclidean ones. We obtain two kinds of results: (1) We show that k-Kemeny remains intractable under most of these domains, even for k=2, and (2) we use k-Kemeny to rank these domains in terms of their diversity.

Problem

• Solving k-Kemeny problem for structured election domains
• Assessing computational intractability across diverse domain types
• Ranking domain diversity using k-Kemeny score metrics

Innovation

• k-Kemeny scores for ranking diversity
• Adjacent candidate swaps to reduce rankings
• Analyzing structured domains for intractability