The Complexity of the Possible Winner Problem over Partitioned Preferences

TL;DR

This paper investigates the computational complexity of the Possible-Winner problem in elections with partitioned preferences, providing polynomial algorithms for certain scoring rules and NP-hardness results for others.

Contribution

It introduces a polynomial-time algorithm for scoring rules with two distinct values and proves NP-hardness for rules with four or more distinct scoring values.

Findings

Polynomial-time algorithm for rules with 2 scoring values

NP-hardness for rules with 4 or more scoring values

Clarifies complexity boundaries for the Possible-Winner problem

Abstract

The Possible-Winner problem asks, given an election where the voters' preferences over the set of candidates is partially specified, whether a distinguished candidate can become a winner. In this work, we consider the computational complexity of Possible-Winner under the assumption that the voter preferences are $partitioned$. That is, we assume that every voter provides a complete order over sets of incomparable candidates (e.g., candidates are ranked by their level of education). We consider elections with partitioned profiles over positional scoring rules, with an unbounded number of candidates, and unweighted voters. Our first result is a polynomial time algorithm for voting rules with $2$ distinct values, which include the well-known $k$-approval voting rule. We then go on to prove NP-hardness for a class of rules that contain all voting rules that produce scoring vectors with at least $4$ distinct values.