We investigate winner determination for two popular proportional representation systems: the Monroe and Chamberlin-Courant (abbrv. CC) systems. Our study focuses on (nearly) single-peaked resp. single-crossing preferences. We show that for single-crossing approval preferences, winner determination of the Monroe rule is polynomial, and for both rules, winner determination mostly admits FPT algorithms with respect to the number of voters to delete to obtain single-peaked or single-crossing preferences. Our results answer some complexity questions from the literature [18, 28, 21].

翻译：我们针对两种常见的比例代表制——Monroe规则和Chamberlin-Courant（简称CC）规则——研究了赢家确定问题。本研究聚焦于（近乎）单峰偏好与单交叉偏好。结果表明，对于单交叉批准偏好，Monroe规则的赢家确定问题是多项式可解的；而对于上述两种规则，在通过删除最少选民以达成单峰或单交叉偏好时，赢家确定问题大多具有关于需删除选民数量的FPT算法。我们的研究结果回答了文献[18, 28, 21]中的若干复杂性疑问。