Kernelization Complexity Of Possible Winner And Coalitional Manipulation Problems In Voting

In the POSSIBLE WINNERproblem 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 WINNERproblem, 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 [ 10]. In this paper, we settle this open question for many common voting rules.We show that the POSSIBLE WINNERproblem 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 WINNERproblem 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 WINNERproblem is harder than the Coalitional Manipulation problem since the Coalitional Manipulation problem admits a polynomial kernel whereas the POSSIBLE WINNERproblem does not admit a polynomial kernel.