Electoral control types are ways of trying to change the outcome of elections by altering aspects of their composition and structure [BTT92]. We say two compatible (i.e., having the same input types) control types that are about the same election system E form a collapsing pair if for every possible input (which typically consists of a candidate set, a vote set, a focus candidate, and sometimes other parameters related to the nature of the attempted alteration), either both or neither of the attempted attacks can be successfully carried out [HHM20]. For each of the seven general (i.e., holding for all election systems) electoral control type collapsing pairs found by Hemaspaandra, Hemaspaandra, and Menton [HHM20] and for each of the additional electoral control type collapsing pairs of Carleton et al. [CCH+ 22] for veto and approval (and many other election systems in light of that paper's Theorems 3.6 and 3.9), both members of the collapsing pair have the same complexity since as sets they are the same set. However, having the same complexity (as sets) is not enough to guarantee that as search problems they have the same complexity. In this paper, we explore the relationships between the search versions of collapsing pairs. For each of the collapsing pairs of Hemaspaandra, Hemaspaandra, and Menton [HHM20] and Carleton et al. [CCH+ 22], we prove that the pair's members' search-version complexities are polynomially related (given access, for cases when the winner problem itself is not in polynomial time, to an oracle for the winner problem). Beyond that, we give efficient reductions that from a solution to one compute a solution to the other. For the concrete systems plurality, veto, and approval, we completely determine which of their (due to our results) polynomially-related collapsing search-problem pairs are polynomial-time computable and which are NP-hard.

翻译：选举控制类型是通过改变选举的构成和结构来试图改变选举结果的方式[BTT92]。对于同一选举系统E，如果两个兼容（即具有相同输入类型）的控制类型构成坍缩对，则对于所有可能的输入（通常包括候选集、投票集、焦点候选者，以及有时与试图改变性质相关的其他参数），要么两种试图攻击都能成功实施，要么都不能成功实施[HHM20]。对于Hemaspaandra、Hemaspaandra和Menton [HHM20]发现的七个通用（即适用于所有选举系统）选举控制类型坍缩对，以及Carleton等人[CCH+ 22]针对否决制和批准制（以及基于该论文定理3.6和3.9的许多其他选举系统）发现的额外选举控制类型坍缩对，坍缩对的两个成员具有相同的复杂度，因为作为集合它们是同一个集合。然而，作为集合具有相同的复杂度并不足以保证作为搜索问题它们也具有相同的复杂度。在本文中，我们探讨了坍缩对的搜索版本之间的关系。对于Hemaspaandra、Hemaspaandra和Menton [HHM20]以及Carleton等人[CCH+ 22]的每个坍缩对，我们证明了该对成员的搜索版本复杂度是多项式相关的（在胜者问题本身不在多项式时间内的情况下，可访问胜者问题的谕示机）。此外，我们给出了高效归约方法，可以从一个问题的解计算出另一个问题的解。对于具体系统如多数制、否决制和批准制，我们完全确定了它们的（基于我们的结果）多项式相关的坍缩搜索问题对中哪些是多项式时间可计算的，哪些是NP困难的。