Kernelization Complexity of Possible Winner and Coalitional Manipulation Problems in Voting

TL;DR

This paper investigates the kernelization complexity of the Possible Winner and Coalitional Manipulation problems in voting, establishing non-existence of polynomial kernels for the former and existence for the latter under certain conditions, highlighting their computational differences.

Contribution

It proves that the Possible Winner problem lacks polynomial kernels for many common voting rules, while the Coalitional Manipulation problem admits polynomial kernels under specific conditions.

Findings

Possible Winner problem has no polynomial kernel for maximin, Copeland, Bucklin, ranked pairs, and certain scoring rules.

Coalitional Manipulation problem admits polynomial kernels when manipulators are polynomial in candidates.

Coalitional Manipulation is computationally easier than Possible Winner in terms of kernelization complexity.

Abstract

In the Possible Winner problem in computational social choice theory, we are given a set of partial preferences and the question is whether a distinguished candidate could be made winner by extending the partial preferences to linear preferences. Previous work has provided, for many common voting rules, fixed parameter tractable algorithms for the Possible Winner problem, with number of candidates as the parameter. However, the corresponding kernelization question is still open and in fact, has been mentioned as a key research challenge. In this paper, we settle this open question for many common voting rules. We show that the Possible Winner problem for maximin, Copeland, Bucklin, ranked pairs, and a class of scoring rules that include the Borda voting rule do not admit a polynomial kernel with the number of candidates as the parameter. We show however that the Coalitional Manipulation problem which is an important special case of the Possible Winner problem does admit a polynomial kernel for maximin, Copeland, ranked pairs, and a class of scoring rules that includes the Borda voting rule, when the number of manipulators is polynomial in the number of candidates. A significant conclusion of our work is that the Possible Winner problem is harder than the Coalitional Manipulation problem since the Coalitional Manipulation problem admits a polynomial kernel whereas the Possible Winner problem does not admit a polynomial kernel.