We focus on a generalization of the classic Minisum approval voting rule, introduced by Barrot and Lang (2016), and referred to as Conditional Minisum (CMS), for multi-issue elections with preferential dependencies. Under this rule, voters are allowed to declare dependencies between different issues, but the price we have to pay for this higher level of expressiveness is that we end up with a computationally hard rule. Motivated by this, we first focus on finding special cases that admit efficient algorithms for CMS. Our main result in this direction is that we identify the condition of bounded treewidth (of an appropriate graph, emerging from the provided ballots) as the necessary and sufficient condition for exact polynomial algorithms, under common complexity assumptions. We then move to the design of approximation algorithms. For the (still hard) case of binary issues, we identify natural restrictions on the voters' ballots, under which we provide the first multiplicative approximation algorithms for the problem. The restrictions involve upper bounds on the number of dependencies an issue can have on the others and on the number of alternatives per issue that a voter can approve. Finally, we also investigate the complexity of problems related to the strategic control of conditional approval elections by adding or deleting either voters or alternatives and we show that in most variants of these problems, CMS is computationally resistant against control. Overall, we conclude that CMS can be viewed as a solution that achieves a satisfactory tradeoff between expressiveness and computational efficiency, when we have a limited number of dependencies among issues, while at the same time exhibiting sufficient resistance to control.

翻译：我们关注Barrot和Lang（2016）提出的经典Minisum批准投票规则的推广形式，即条件最小和规则（CMS），用于存在偏好依赖的多议题选举。在该规则下，选民被允许声明不同议题之间的依赖关系，但为此更高表达力付出的代价是规则的计算难度增加。基于此，我们首先聚焦于寻找允许对CMS实现高效算法的特殊情形。在这一方向上的主要成果是：在一般复杂性假设下，我们识别出有界树宽（通过提交选票构建的特定图结构）作为精确多项式算法的充要条件。随后转向近似算法设计。针对（仍然困难的）二元议题情形，我们识别出选民选票中的自然约束条件，在此条件下首次提出问题的乘法近似算法。这些约束包括议题间依赖关系数量的上界，以及每个议题中选民可批准的备选方案数量上界。最后，我们还研究了通过添加或删除选民或备选方案对条件批准选举进行策略控制的相关问题的复杂性，并证明在大多数变体中，CMS对控制具有计算抵抗性。总体而言，我们得出结论：当议题间依赖关系数量有限时，CMS可视为在表达力和计算效率之间实现满意平衡的方案，同时表现出充分的抗控制性。