Voting in Two-Crossing Elections

TL;DR

This paper introduces two-crossing elections as a generalization of single-crossing elections, providing polynomial recognition algorithms, complexity results, and efficient algorithms for certain voting rules, highlighting differences in computational complexity.

Contribution

It establishes polynomial-time recognition for two-crossing elections, explores their properties, and analyzes the complexity of voting rules within this framework, contrasting with single-crossing elections.

Findings

Two-crossing elections can be recognized in polynomial time.

Many voting rules are NP-hard under two-crossing elections.

Young scores and Chamberlin-Courant rule can be computed efficiently in two-crossing elections.

Abstract

We introduce two-crossing elections as a generalization of single-crossing elections, showing a number of new results. First, we show that two-crossing elections can be recognized in polynomial time, by reduction to the well-studied consecutive ones problem. We also conjecture that recognizing $k$-crossing elections is NP-complete in general, providing evidence by relating to a problem similar to consecutive ones proven to be hard in the literature. Single-crossing elections exhibit a transitive majority relation, from which many important results follow. On the other hand, we show that the classical Debord-McGarvey theorem can still be proven two-crossing, implying that any weighted majority tournament is inducible by a two-crossing election. This shows that many voting rules are NP-hard under two-crossing elections, including Kemeny and Slater. This is in contrast to the single-crossing case and outlines an important complexity boundary between single- and two-crossing. Subsequently, we show that for two-crossing elections the Young scores of all candidates can be computed in polynomial time, by formulating a totally unimodular linear program. Finally, we consider the Chamberlin-Courant rule with arbitrary disutilities and show that a winning committee can be computed in polynomial time, using an approach based on dynamic programming.