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A heuristic search algorithm for discovering large Condorcet domains

Bei Zhou, Søren Riis

TL;DR

This work targets the open problem of identifying large Condorcet domains for fixed numbers of alternatives. It introduces a universal heuristic search that leverages a database of five-alternative restriction patterns to predict the potential size of complete CDs, guiding exploration via a linear, size-correlated scoring function and a stable triple ordering. The approach yields new record CDs for n=10 (size 1082) and n=11 (size 2349), while shedding light on the CD structure through Ramsey-theoretic observations and restriction-pattern analyses. Although the method hinges on a database with scale limitations and non-local effects for larger n, it provides a scalable, parallelizable framework that establishes new benchmarks and informs future theoretical and algorithmic work in Condorcet-domain discovery.

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.

A heuristic search algorithm for discovering large Condorcet domains

TL;DR

This work targets the open problem of identifying large Condorcet domains for fixed numbers of alternatives. It introduces a universal heuristic search that leverages a database of five-alternative restriction patterns to predict the potential size of complete CDs, guiding exploration via a linear, size-correlated scoring function and a stable triple ordering. The approach yields new record CDs for n=10 (size 1082) and n=11 (size 2349), while shedding light on the CD structure through Ramsey-theoretic observations and restriction-pattern analyses. Although the method hinges on a database with scale limitations and non-local effects for larger n, it provides a scalable, parallelizable framework that establishes new benchmarks and informs future theoretical and algorithmic work in Condorcet-domain discovery.

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.
Paper Structure (10 sections, 2 theorems, 3 equations, 3 figures, 14 tables, 2 algorithms)

This paper contains 10 sections, 2 theorems, 3 equations, 3 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and background
  3. Suboptimality of existing learning algorithms
  4. The algorithm
  5. Database construction
  6. The heuristic function
  7. The heuristic search algorithm
  8. Experiments
  9. Analysis on discovered new large CDs
  10. Conclusion

Key Result

Corollary 1

For any positive integer $k$, there exists a positive integer $n (n>k)$, such that any CD on $n$ alternatives has a restriction to $k$ alternatives where the rules on them are the same.

Figures (3)

  • Figure 1: The restriction sizes for the Fishburn domain on five alternatives when restricted to 4 alternatives.
  • Figure 2: The size of 5 restrictions on 4 alternatives from a five alternative CD of size 17.
  • Figure 3: The correlation between the sizes of domains and their values evaluated by the value function for six and seven alternative CDs. Plot (a) includes all the six alternative isomorphic CDs; plot (b) uses 100000 randomly sampled $n=7$ CDs for each size if the number of CDs is too many.

Theorems & Definitions (2)

  • Corollary
  • Corollary