We explore large peak-pit Condorcet domains (CD), an active research area in voting theory. The search for large CDs, defined combinatorially, serves as a benchmark for heuristic-based optimisation algorithms. Since 1996, Fishburn’s alternating scheme produced the largest known CDs for \(n \le 15\) alternatives until recent discoveries surpassed it for \(n = 8\) and \(n \ge 13\) . For \(8< n < 13\) , exhaustive searches are infeasible, necessitating the design of heuristic methods. We developed a novel algorithm using a specially designed heuristic function and a 5-alternative subset lookup database to obtain new record-size CDs in this critical range. This approach, distinct from existing methods used for \(n \le 8\) , found new large peak-pit CDs: size 1082 (previously 1069) for \(n = 10\) , and 2349 (previously 2324) for \(n = 11\) , establishing new lower bounds for these cases. Notably, these new CDs hold restrictions of remarkably small size. These findings fill a significant gap in our knowledge of CDs on \(8< n < 13\) and offer new insights into the structure of large CDs.
