Search versus Search for Collapsing Electoral Control Types

For each of the seven general electoral control type collapses found by Hemaspaandra, Hemaspaandra, and Menton [HHM20] and each of the additional electoral control type collapses 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), the collapsing types obviously 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' 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. We also, for the concrete systems plurality, veto, and approval completely determine which of those polynomially-related search-problem pairs are polynomial-time computable and which are NP-hard.