A heuristic search algorithm for discovering large Condorcet domains

TL;DR

This paper introduces a heuristic search algorithm that discovers larger Condorcet domains for specific numbers of options, surpassing previous records and revealing new structural characteristics.

Contribution

The paper presents a novel heuristic search method with a lookup-backed heuristic function that finds larger Condorcet domains than previously known.

Findings

Discovered a Condorcet domain of size 1082 for n=10

Found a Condorcet domain of size 2349 for n=11

New CDs exhibit distinct structural characteristics

Abstract

The study of large Condorcet domains (CD) has been a significant area of interest in voting theory. In this paper, our goal is to search for large CDs that are hitherto unknown. With a straightforward combinatorial definition, searching for large CDs is naturally suited for algorithmic optimisations. For each value of n>2, one can ask for the size of the largest CD, thus finding the largest CDs provides an important benchmark for heuristic-based combinatorial optimisation algorithms. Despite extensive research over the past three decades, the CD sizes identified in 1996 remain the best known for many values of n. When n>8, conducting an exhaustive search becomes computationally unfeasible, thereby prompting the use of heuristic methods. To address this, we developed a novel heuristic search algorithm in which a specially designed heuristic function, backed by a lookup database, directs the search towards promising branches in the search tree. Our algorithm found new large CDs of size 1082 (surpassing the previous record of 1069) for n=10, and 2349 (improving the previous 2324) for n=11. Notably, these newly discovered CDs exhibit characteristics distinct from those of known CDs.