Recovering Single-Crossing Preferences From Approval Ballots

TL;DR

This paper presents a polynomial-time algorithm to determine if a set of approval ballots can be derived from a single-crossing preference profile, addressing an open problem in computational social choice.

Contribution

It introduces the first efficient method to identify single-crossing preferences from approval ballots and explores the complexity of characterizing negative instances.

Findings

Polynomial-time algorithm for single-crossing preferences detection

Existence of forbidden sub-ballots characterization requires infinitely many configurations

Addresses an open problem in computational social choice

Abstract

An electorate with fully-ranked innate preferences casts approval votes over a finite set of alternatives. As a result, only partial information about the true preferences is revealed to the voting authorities. In an effort to understand the nature of the true preferences given only partial information, one might ask whether the unknown innate preferences could possibly be single-crossing. The existence of a polynomial time algorithm to determine this has been asked as an outstanding problem in the works of Elkind and Lackner. We hereby give a polynomial time algorithm determining a single-crossing collection of fully-ranked preferences that could have induced the elicited approval ballots, or reporting the nonexistence thereof. Moreover, we consider the problem of identifying negative instances with a set of forbidden sub-ballots, showing that any such characterization requires infinitely many forbidden configurations.