On Problem Kernels for Possible Winner Determination under the k-Approval Protocol

The POSSIBLE WINNER problem asks whether some distinguished candidate may become the winner of an election when the given incomplete votes (partial orders) are extended into complete ones (linear orders) in a favorable way. Under the k-approval protocol, for every voter, the best k candidates of his or her preference order get one point. A candidate with maximum total number of points wins. The POSSIBLE WINNER problem for k-approval is NP-complete even if there are only two votes (and k is part of the input). In addition, it is NPcomplete for every fixed k ∈ {2, . . . , m − 2} with m denoting the number of candidates if the number of votes is unbounded. We investigate the parameterized complexity with respect to the combined parameter k and “number of incomplete votes” t, and with respect to the combined parameter k′ := m − k and t. For both cases, we use kernelization to show fixed-parameter tractability. However, we show that whereas there is a polynomial-size problem kernel with respect to (t, k′), it is very unlikely that there is a polynomial-size kernel for (t, k). We provide additional fixed-parameter algorithms for some special cases.