Towards a Dichotomy for the Possible Winner Problem in Elections Based on Scoring Rules

TL;DR

This paper investigates the computational complexity of the Possible Winner problem in elections using scoring rules, establishing NP-completeness for most rules and polynomial solvability for some, thus clarifying the problem's difficulty landscape.

Contribution

It provides a comprehensive complexity classification of the Possible Winner problem for all scoring rules except one, extending previous partial results.

Findings

Possible Winner is NP-complete for most scoring rules with unbounded candidates and voters.

Polynomial-time solutions exist for plurality and veto rules.

The paper settles the complexity for nearly all scoring rules except one.

Abstract

To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the Possible Winner problem that asks, given a set of partial votes, whether a distinguished candidate can still become a winner. In this work, we consider the computational complexity of Possible Winner for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, k-approval, and Borda. Generalizing previous NP-hardness results for some special cases, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that Possible Winner is NP-complete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2,1,...,1,0), while it is solvable in polynomial time for plurality and veto.