We study the computational complexity of bribery in parliamentary voting, in settings where the briber is (also) interested in the success of an entire set of political parties - a ``coalition'' - rather than an individual party. We introduce two variants of the problem: the Coalition-Bribery Problem (CB) and the Coalition-Bribery-with-Preferred-party Problem (CBP). In CB, the goal is to maximize the total number of seats held by a coalition, while in CBP, there are two objectives: to maximize the votes for the preferred party, while also ensuring that the total number of seats held by the coalition is above the target support (e.g. majority). We study the complexity of these bribery problems under two positional scoring functions - Plurality and Borda - and for multiple bribery types - $1$-bribery, $\$$-bribery, swap-bribery, and coalition-shift-bribery. We also consider both the case where seats are only allotted to parties whose number of votes passes some minimum support level and the case with no such minimum. We provide polynomial-time algorithms to solve some of these problems and prove that the others are NP-hard.

翻译：本研究探讨了议会投票中贿赂问题的计算复杂性，其中行贿者不仅关注单个政党的成功，更关注整个政治党派集合——即“联盟”——的整体利益。我们引入了该问题的两种变体：联盟贿赂问题（CB）与含首选政党的联盟贿赂问题（CBP）。在CB问题中，目标是最大化联盟持有的总席位数量；而在CBP问题中，存在双重目标：既要为首选政党争取最多选票，又要确保联盟持有的总席位数量达到目标支持度（例如多数席位）。我们研究了在两种位置计分函数——多数制（Plurality）与波达计数法（Borda）——以及多种贿赂类型（$1$-贿赂、$\$$-贿赂、交换贿赂与联盟转移贿赂）下这些贿赂问题的复杂性。同时考虑了两种席位分配情形：仅将席位分配给得票数超过最低支持门槛的政党，以及无此门槛的情形。我们针对部分问题提出了多项式时间算法，并证明了其余问题具有NP难度。