Elections with Few Voters: Candidate Control Can Be Easy

TL;DR

This paper explores how the complexity of candidate control in elections varies with the number of voters, revealing diverse computational challenges across different voting rules and scenarios.

Contribution

It provides a detailed analysis of the parameterized complexity of candidate control in elections with few voters, highlighting differences from elections with many voters.

Findings

Complexity varies significantly across voting rules.

Adding or deleting candidates can be computationally easy or hard.

Results differ between standard and combinatorial scenarios.

Abstract

We study the computational complexity of candidate control in elections with few voters, that is, we consider the parameterized complexity of candidate control in elections with respect to the number of voters as a parameter. We consider both the standard scenario of adding and deleting candidates, where one asks whether a given candidate can become a winner (or, in the destructive case, can be precluded from winning) by adding or deleting few candidates, as well as a combinatorial scenario where adding/deleting a candidate automatically means adding or deleting a whole group of candidates. Considering several fundamental voting rules, our results show that the parameterized complexity of candidate control, with the number of voters as the parameter, is much more varied than in the setting with many voters.