Are there any nicely structured preference~profiles~nearby?

TL;DR

This paper studies the computational complexity of determining whether a preference profile can be made to exhibit certain desirable structures by removing a small number of voters or alternatives, classifying which cases are tractable or NP-hard.

Contribution

It provides a comprehensive complexity classification for the problem of editing preference profiles to achieve structured forms by deletions.

Findings

Some profile structures can be achieved efficiently via polynomial-time algorithms.

Other structures are NP-hard to obtain through minimal deletions.

The paper delineates the boundary between tractable and intractable cases.

Abstract

We investigate the problem of deciding whether a given preference profile is close to having a certain nice structure, as for instance single-peaked, single-caved, single-crossing, value-restricted, best-restricted, worst-restricted, medium-restricted, or group-separable profiles. We measure this distance by the number of voters or alternatives that have to be deleted to make the profile a nicely structured one. Our results classify the problem variants with respect to their computational complexity, and draw a clear line between computationally tractable (polynomial-time solvable) and computationally intractable (NP-hard) questions.