We study the Possible President problem and the Necessary President problem for Schulze voting, a rule that, due to its many desirable axiomatic properties, is popular in practice. In both problems, we are given an election with the candidates partitioned into a set of parties, and we are interested in questions about a given distinguished party. In the Possible President problem, we ask whether it is possible for the parties to each nominate exactly one candidate such that the nominee of the distinguished party is a Schulze winner of the resulting election with only the nominees running. In the Necessary President problem, we ask whether the distinguished party's nominee is a Schulze winner of the resulting election, irrespective of the nomination from the other parties. Rothe and Woitaschik have shown that Possible President is NP-complete and Necessary President is coNP-complete for Schulze elections. We complement and improve their results by a more fine-grained analysis: we determine the parameterized complexity of both problems with respect to all possible parameterizations, where we consider each of three natural parameters -- the number of voters, the maximum party size, and the number of parties -- to be either a constant, a parameter, or unbounded. In particular, we obtain dichotomies regarding the number of voters for both problems.

翻译：我们研究了舒尔茨投票中的可能总统问题与必然总统问题。舒尔茨规则因其诸多理想公理性质而在实践中广受欢迎。在这两个问题中，我们给定一个选举，其中候选人被划分为若干政党集合，并关注某个特定政党。在可能总统问题中，我们询问是否存在一种方案，使得每个政党恰好提名一名候选人，且该特定政党的提名者在仅由提名者构成的最终选举中成为舒尔茨赢家。在必然总统问题中，我们询问无论其他政党如何提名，该特定政党的提名者是否总能成为最终选举的舒尔茨赢家。Rothe和Woitaschik已证明，对于舒尔茨选举，可能总统问题是NP完全的，必然总统问题是coNP完全的。我们通过更精细的分析补充并改进了他们的结果：针对所有可能的参数化方式，我们确定了这两个问题的参数化复杂性——将三个自然参数（选民数量、最大政党规模、政党数量）分别视为常数、参数或无界。特别地，我们获得了关于选民数量对两个问题的二分法结果。