Heuristic Search Algorithm Discovers New Record-Breaking Condorcet Domains on 10 and 11 Alternatives

Temel Kavramlar

A novel heuristic search algorithm utilizing a lookup database and a specially designed heuristic function has discovered new record-breaking Condorcet domains of size 1082 on 10 alternatives and 2349 on 11 alternatives, surpassing the previous best-known sizes.

Özet

The paper presents a breakthrough in the study of large Condorcet domains (CDs), which are collections of linear orders (permutations) where every triple of candidates satisfies a specific "never" condition, ensuring that cyclical majorities are avoided.
The key highlights and insights are:

The authors identify significant challenges faced by common learning algorithms, such as reinforcement learning, genetic algorithms, and local search, in finding large CDs, especially for a substantial number of alternatives. These challenges include the exponential growth of the search space, the presence of many local optima, and the computational complexity of evaluating CD sizes.

To address these challenges, the authors develop a novel heuristic search algorithm that employs an efficient heuristic function. This function evaluates the goodness of partial CDs based on the sizes of their restricted subset domains, exploiting the empirical finding that many locally large restricted CDs tend to be large.

The heuristic function utilizes a pre-calculated database containing information on all possible CDs on five alternatives, resonating with the concept of dynamic programming where subproblems are pre-computed and reused.

The search algorithm combines the merits of reinforcement learning, evolutionary algorithms, and local search, coalescing techniques to effectively navigate the vast search space and avoid local optima.

The authors' search algorithm discovered new record-breaking CDs of size 1082 on 10 alternatives and 2349 on 11 alternatives, surpassing the previous best-known sizes. Notably, these newly discovered CDs exhibit characteristics distinct from the known Fishburn domains, challenging the existing paradigm.

The authors provide a detailed analysis of the structures and features of the new large CDs, comparing them to the Fishburn domains and presenting the list of rules leading to the largest CDs.

İstatistikler

The largest known Condorcet domains for 4 to 13 alternatives are:
4 alternatives: 9 permutations
5 alternatives: 20 permutations
6 alternatives: 45 permutations
7 alternatives: 100 permutations
8 alternatives: 224 permutations
9 alternatives: 488 permutations
10 alternatives: 1082 permutations (new record)
11 alternatives: 2349 permutations (new record)
12 alternatives: 5034 permutations
13 alternatives: 10840 permutations

Alıntılar

"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."

Önemli Bilgiler Şuradan Elde Edildi

A heuristic search algorithm for discovering large Condorcet domains

by Bei ... : arxiv.org 04-29-2024

https://arxiv.org/pdf/2303.06524.pdf

Daha Derin Sorular

How can the heuristic function be further improved to establish an even stronger linear relationship between the size of a Condorcet domain and its evaluated value?

To further enhance the heuristic function and establish a stronger linear relationship between the size of a Condorcet domain and its evaluated value, several strategies can be considered:

Feature Engineering: Introduce additional features or metrics that capture more nuanced characteristics of the Condorcet domains. These features could include properties related to the structure, symmetry, or connectivity of the domain. By incorporating more relevant information into the heuristic function, a more comprehensive evaluation can be achieved.

Non-linear Transformations: Explore the possibility of incorporating non-linear transformations or interactions between features in the heuristic function. Non-linear relationships between the size of a domain and its value may exist, and capturing these relationships can lead to a more accurate evaluation.

Regularization Techniques: Implement regularization techniques to prevent overfitting and ensure that the heuristic function generalizes well to unseen data. Regularization methods such as L1 or L2 regularization can help in controlling the complexity of the function and improving its predictive performance.

Ensemble Methods: Utilize ensemble learning techniques to combine multiple heuristic functions or models. By aggregating the predictions of diverse models, the overall performance of the heuristic function can be enhanced, leading to a more robust and accurate evaluation of Condorcet domains.

Hyperparameter Tuning: Conduct thorough hyperparameter tuning to optimize the parameters of the heuristic function. Fine-tuning the weights, thresholds, or other parameters in the function can help in achieving a better alignment between the evaluated values and the actual sizes of the domains.

By iteratively refining the heuristic function using these strategies and potentially exploring advanced machine learning approaches, it is possible to establish a more robust and precise linear relationship between the size of a Condorcet domain and its evaluated value.

What are the potential implications of the newly discovered Condorcet domains that do not follow the Fishburn alternating scheme, and how might they impact the theoretical study of voting systems?

The discovery of Condorcet domains that do not adhere to the Fishburn alternating scheme carries several significant implications and impacts on the theoretical study of voting systems:

Diversification of Domain Structures: The existence of non-Fishburn domains expands the diversity of possible domain structures, challenging the conventional understanding that maximal Condorcet domains are solely constructed based on the alternating scheme. This diversification prompts a reevaluation of the fundamental principles underlying Condorcet domains.

Theoretical Framework Enrichment: The discovery of these new domains enriches the theoretical framework of voting systems by introducing novel structures and patterns that deviate from traditional paradigms. This opens up avenues for exploring alternative approaches to preference aggregation and decision-making processes.

Algorithmic Development: The identification of Condorcet domains outside the Fishburn scheme may inspire the development of new algorithmic techniques and optimization strategies for discovering and analyzing complex domain structures. This can lead to advancements in computational methods for studying voting theory.

Impact on Voting System Design: The presence of non-Fishburn domains may influence the design and evaluation of voting systems. By considering a broader range of domain structures, researchers and policymakers can gain insights into the implications of different voting mechanisms on preference aggregation and democratic decision-making.

Future Research Directions: The exploration of these new Condorcet domains sets the stage for further research into the properties, characteristics, and implications of diverse domain structures. Future studies may focus on understanding the factors influencing the formation of these domains and their implications for voting system design and analysis.

Overall, the discovery of Condorcet domains that do not conform to the Fishburn alternating scheme enriches the theoretical landscape of voting systems, stimulates further research inquiries, and broadens the understanding of preference aggregation mechanisms.

Could the techniques and insights developed in this work be applied to solve other complex combinatorial optimization problems beyond Condorcet domain discovery?

The techniques and insights developed in the study of Condorcet domain discovery hold potential for application in solving a variety of other complex combinatorial optimization problems. Some ways in which these techniques could be extended to address broader optimization challenges include:

Heuristic Function Design: The heuristic function developed for evaluating Condorcet domains can be adapted and customized to assess solutions in different combinatorial optimization problems. By defining relevant features and metrics specific to the problem domain, a tailored heuristic function can effectively guide the search process.

Search Algorithm Optimization: The search algorithm employed for exploring Condorcet domains can be modified and optimized for other combinatorial optimization tasks. Techniques such as tree traversal, node expansion, and solution selection can be adapted to suit the characteristics of different problem domains.

Database Utilization: The use of a precomputed database to store and retrieve information about partial solutions can be leveraged in other optimization problems. By maintaining a repository of relevant data and solutions, the efficiency and effectiveness of the search process can be enhanced.

Machine Learning Integration: The integration of machine learning techniques, such as reinforcement learning or genetic algorithms, can further improve the search and optimization process for a wide range of combinatorial problems. By incorporating learning algorithms, the system can adapt and evolve based on feedback and experience.

Generalization to Various Domains: The fundamental principles and methodologies developed in the study of Condorcet domains can be generalized and applied to diverse combinatorial optimization domains, including scheduling problems, network optimization, and resource allocation. By abstracting the core concepts, the techniques can be extended to address different optimization challenges.

In conclusion, the techniques and insights derived from the study of Condorcet domain discovery possess versatility and applicability beyond their original context. By adapting and extending these methodologies, researchers can tackle a wide array of complex combinatorial optimization problems with enhanced efficiency and effectiveness.