Consider an election where the set of candidates is partitioned into parties, and each party must choose exactly one candidate to nominate for the election held over all nominees. The Necessary President problem asks whether a candidate, if nominated, becomes the winner of the election for all possible nominations from other parties. We study the computational complexity of Necessary President for several voting rules. We show that while this problem is solvable in polynomial time for Borda, Maximin, and Copeland$^α$ for every $α\in [0,1]$, it is $\mathsf{coNP}$-complete for general classes of positional scoring rules that include $\ell$-Approval and $\ell$-Veto, even when the maximum size of a party is two. For such positional scoring rules, we show that Necessary President is $\mathsf{W}[2]$-hard when parameterized by the number of parties, but fixed-parameter tractable with respect to the number of voter types. Additionally, we prove that Necessary President for Ranked Pairs is $\mathsf{coNP}$-complete even for maximum party size two, and $\mathsf{W}[1]$-hard with respect to the number of parties; remarkably, both of these results hold even for constant number of voters.

翻译：考虑一种选举，其中候选人集合被划分为若干政党，每个政党必须恰好选择一名候选人提名，并在所有提名候选人中进行选举。必然当选者问题询问：若某位候选人被提名，是否在所有其他政党可能的提名组合下，该候选人都会成为选举的获胜者。我们研究了必然当选者问题在多种投票规则下的计算复杂性。结果表明，对于Borda、Maximin以及任意$α\in [0,1]$的Copeland$^α$规则，该问题可在多项式时间内求解；然而对于包含$\ell$-Approval和$\ell$-Veto在内的广义位置计分规则类，即使政党最大规模为二，该问题也是$\mathsf{coNP}$完全的。针对此类位置计分规则，我们证明必然当选者问题以政党数量为参数时是$\mathsf{W}[2]$难的，但以选民类型数量为参数时是固定参数可解的。此外，我们证明了Ranked Pairs规则下的必然当选者问题即使在政党最大规模为二时也是$\mathsf{coNP}$完全的，且以政党数量为参数是$\mathsf{W}[1]$难的；值得注意的是，这两个结论在选民数量恒定时依然成立。