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：选举控制类型是指通过改变选举的组成和结构来试图改变选举结果的方式[BTT92]。若两个兼容（即输入类型相同）且针对同一选举系统E的控制类型构成一个“坍塌对”，则对于每个可能的输入（通常包括候选集、投票集、焦点候选人，以及有时与尝试篡改性质相关的其他参数），要么两种攻击都能成功实施，要么均不能成功实施[HHM20]。对于Hemaspaandra、Hemaspaandra和Menton[HHM20]发现的七组通用（即适用于所有选举系统）选举控制类型坍塌对，以及Carleton等人[CCH+ 22]针对否决制和批准制（结合该文定理3.6和3.9，也适用于许多其他选举系统）发现的额外选举控制类型坍塌对，由于坍塌对的两个成员在集合意义上完全相同，因此它们具有相同的复杂性。然而，作为集合具有相同复杂性并不足以保证它们作为搜索问题也具有相同复杂性。本文探索了搜索版本中坍塌对之间的关系。对于Hemaspaandra、Hemaspaandra和Menton[HHM20]以及Carleton等人[CCH+ 22]提出的每个坍塌对，我们证明其成员的搜索版本复杂性是多项式相关的（在胜者问题本身不属于多项式时间类的情况下，可访问胜者问题的谕示机）。除此之外，我们给出了高效归约方法，能从其中一个问题的解计算出另一个问题的解。针对具体的多元制、否决制和批准制系统，我们完全确定了哪些（基于我们的结果而）多项式相关的坍塌搜索问题对是多项式时间可计算的，哪些是NP难的。