How Hard Is It for a Party to Nominate an Election Winner?

We consider a Plurality-voting scenario, where the candidates are split between parties, and each party nominates exactly one candidate for the final election. We study the computational complexity of deciding if there is a set of nominees such that a candidate from a given party wins in the final election. In our second problem, the goal is to decide if a candidate from a given party always wins, irrespective who is nominated. We show that these problems are computationally hard, but are polynomial-time solvable for restricted settings.