Single-crossing Implementation

TL;DR

This paper introduces a graph-based approach to analyze elections for single-crossing properties, enabling the detection of elections close to being single-crossing through minimal modifications.

Contribution

It proposes a novel graph-theoretic mapping to identify obstacles preventing elections from being single-crossing and studies the complexity of detecting nearly single-crossing elections.

Findings

Graph mapping effectively encodes obstacles to single-crossing

Detecting nearly single-crossing elections is computationally analyzed

The approach facilitates understanding of election structure and complexity

Abstract

An election over a finite set of candidates is called single-crossing if, as we sweep through the list of voters from left to right, the relative order of every pair of candidates changes at most once. Such elections have many attractive properties: e.g., their majority relation is transitive and they admit efficient algorithms for problems that are NP-hard in general. If a given election is not single-crossing, it is important to understand what are the obstacles that prevent it from having this property. In this paper, we propose a mapping between elections and graphs that provides us with a convenient encoding of such obstacles. This mapping enables us to use the toolbox of graph theory in order to analyze the complexity of detecting nearly single-crossing elections, i.e., elections that can be made single-crossing by a small number of modifications.