New Record-Breaking Condorcet Domains on 10 and 11 Alternatives

The study of large Condorcet domains (CD) has been a significant area of interest in voting theory. In this paper, our goal was to search for large CDs. With a straightforward combinatorial definition, searching for large CDs is naturally suited for algorithmic techniques and optimisations. For each value of $n>2$, one can ask for the largest CD, and we suggest that finding new record-sized 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 introduce a novel heuristic search algorithm in which a specially designed heuristic function, backed by a lookup database, directs the search towards beneficial branches in the search tree. We report new records of sizes 1082 (surpassing the previous record of 1069) for $n=10$, and 2349 (improving the previous 2324) for $n=11$. Notably, these discoveries exhibit characteristics distinct from those of known CDs. We examine the structure of these newfound Condorcet domains. Our results underscore the potential of AI-driven and inspired techniques in pushing the boundaries of mathematical research and tackling challenging combinatorial optimisation tasks.