Hardness of candidate nomination

We consider elections where the set of candidates is split into parties and each party can nominate just one candidate. We study the computational complexity of two problems. The Possible President problem asks whether a given party candidate can become the unique winner of the election for some nominations from other parties. The Necessary President is the problem to decide whether a given candidate will be the unique winner of the election for any possible nominations from other parties. We consider several different voting rules and show that for all of them the Possible President problem is NP-complete, even if the size of each party is at most two; for some voting rules we prove that the Necessary President is coNP-complete. Further, we formulate integer programs to solve the Possible President and Necessary President problems and test them on real and artificial data.