An efficient heuristic search algorithm for discovering large Condorcet domains
File(s)
Published version
Author(s)
Zhou, Bei
Riis, Søren
Type
Journal Article
Abstract
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 ≤ 15 alternatives until recent discoveries
surpassed it for n = 8 and n ≥ 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 ≤ 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.
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 ≤ 15 alternatives until recent discoveries
surpassed it for n = 8 and n ≥ 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 ≤ 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.
Date Issued
2025-06-01
Date Acceptance
2024-12-28
Citation
4OR, 2025, 23 (2), pp.193-216
ISSN
1619-4500
Publisher
Springer Science and Business Media LLC
Start Page
193
End Page
216
Journal / Book Title
4OR
Volume
23
Issue
2
Copyright Statement
© The Author(s) 2025 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Publication Status
Published
Date Publish Online
2025-01-24