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Efficient space reduction techniques by optimized majority rules for the Kemeny aggregation problem and beyond

Xuan Kien Phung, Sylvie Hamel

Abstract

The Kemeny aggregation problem consists of computing the consensus rankings of an election with respect to the well-known Kemeny-Young voting method. These consensus rankings satisfy various fundamental properties and are the geometric medians of the votes in the election under the Kendall-tau distance which counts the number of pairwise disagreements. The Kemeny aggregation problem admits important applications in various domains such as computational social choice, machine learning, operations research, and biology but it is unfortunately NP-hard. Recently, Milosz and the second author presented an approach to reduce the search space of the problem by solving the relative order of pairs of elements in those consensus. In this article, we prove an optimized extension of this approach achieving significantly more refined space reduction techniques without adding much to the running time of the algorithms in practice, as illustrated by experimental results and analysis on real and synthetic data. We show how the constraints built by our approach can be used in combination with other methods such as Integer Programming and Finest Condorcet Partitioning to achieve an efficient and scalable solution approach to the Kemeny aggregation problem. Relaxed and approximate versions of our algorithms are also described and evaluated. We also provide practical methods to compute provable guarantees for the quality of the approximate rankings obtained.

Efficient space reduction techniques by optimized majority rules for the Kemeny aggregation problem and beyond

Abstract

The Kemeny aggregation problem consists of computing the consensus rankings of an election with respect to the well-known Kemeny-Young voting method. These consensus rankings satisfy various fundamental properties and are the geometric medians of the votes in the election under the Kendall-tau distance which counts the number of pairwise disagreements. The Kemeny aggregation problem admits important applications in various domains such as computational social choice, machine learning, operations research, and biology but it is unfortunately NP-hard. Recently, Milosz and the second author presented an approach to reduce the search space of the problem by solving the relative order of pairs of elements in those consensus. In this article, we prove an optimized extension of this approach achieving significantly more refined space reduction techniques without adding much to the running time of the algorithms in practice, as illustrated by experimental results and analysis on real and synthetic data. We show how the constraints built by our approach can be used in combination with other methods such as Integer Programming and Finest Condorcet Partitioning to achieve an efficient and scalable solution approach to the Kemeny aggregation problem. Relaxed and approximate versions of our algorithms are also described and evaluated. We also provide practical methods to compute provable guarantees for the quality of the approximate rankings obtained.

Paper Structure

This paper contains 18 sections, 11 theorems, 18 equations, 5 figures, 6 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. Backgrounds and notations
  3. The Major Order Theorem (MOT)
  4. Extended Condorcet criterion (XCC) and approximate Condorcet partition (ACP)
  5. Optimized MOT: $\pmb \alpha$MOT
  6. Sequential algorithms for $\pmb \alpha$MOT
  7. Parallel algorithms for $\pmb \alpha$MOT
  8. More constraints in some medians: $\pmb \alpha$MOT$\text{e}$ and greedy variants G1, G2
  9. $\pmb \alpha$MOTe
  10. $\pmb \alpha$MOTe-Greedy and $\pmb {G1}$, $\pmb{G2}$
  11. Solving pairs with low interference
  12. $M$-finest Condorcet partitions
  13. Refined Approximate Condorcet Partitions by variants of $\pmb \alpha$MOT
  14. Local optimization for approximate Kemeny solutions and applications
  15. Experiments
  16. ...and 3 more sections

Key Result

Theorem 1

Let $(C,V)$ be an election. Let $W_{-1}=\varnothing$ and for every $k \geq 0$, we define $W_k$ inductively as the set of ordered pairs $(x,y)$ of distinct candidates such that where the interference set $Z_k(x,y)$ is defined by Suppose that $(x,y) \in \overline{W}_{k}$ for some $k \geq 0$. Then $x$ is ranked before $y$ in every median of the election.

Figures (5)

  • Figure 1: The scheme representing the main methods of the paper and their interactions.
  • Figure 2: Digraph showing all 13 ordered pairs solved by MOT (black) and the extra 11 ordered pairs solved by $\alpha$ MOT (red) in all medians of the election given in Example \ref{['ex:toy-example-1']}. The extra 4 ordered pairs solved by $\alpha$MOTe (green) are valid for some median of the same election.
  • Figure 3: Comparisons of the size of the set of constraints obtained by $\alpha$MOT and its variants.
  • Figure 4: Digraph showing all 28 ordered pairs found by the greedy method G2 for the election given in Example \ref{['ex:toy-example-1']}. The blue edges correspond to ordered pairs found by G2 which contradict all the medians of the election.
  • Figure 5: Number of correctly solved instances by $M$-ACP methods on 100 random Mallows voting profiles with 50 candidates, $m\in \{3,4,5,9,10,11,14,15,16,19,20,21\}$ votes, dispersion parameter $\theta=0.9$, interference threshold $r=10$, and approximation threshold $h=24$.

Theorems & Definitions (13)

  • Theorem 1: $\alpha$MOT
  • Theorem 2
  • Theorem 3: $\alpha$MOTe
  • Definition 1
  • Definition 2
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Theorem 7
  • Theorem 8
  • ...and 3 more