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Electoral control refers to attacking elections by adding, deleting, or partitioning voters or candidates [3]. Hemaspaandra et al. [16] discovered, for seven pairs (T , T ′ ) of seemingly distinct standard electoral control types, that T and T ′ are identical: For each input 𝐼 and each election system E, 𝐼 is a “yes” instance of both T and T ′ under E, or of neither. Surprisingly, this had gone undetected even as the field was scorecarding how many standard control types election systems were resistant to; various “different” cells on such score cards were, unknowingly, duplicate effort on the same issue. This naturally raises the worry that other pairs of control types are also identical, and so work still is being needlessly duplicated. We determine, for all standard control types, which pairs are, for elections whose votes are linear orderings of the candidates, always identical. We show that no identical control pairs exist beyond the known seven. For three central election systems, we determine which control pairs are identical (“collapse”) with respect to those particular systems, and we explore containment/incomparability relationships between control pairs. For approval voting, which has a different “type” for its votes, Hemaspaandra et al.’s [16] seven collapses still hold. But we find 14 additional collapses that hold for approval voting but not for some election systems whose votes are linear orderings. We find one additional collapse for veto and none for plurality. We prove that each of the three election sys tems mentioned have no collapses other than those inherited from Hemaspaandra et al. [16] or added here. But we show many new containment relationships that hold between some separating con trol pairs, and for each separating pair of standard control types classify its separation in terms of containment (always, and strict on some inputs) or incomparability. Our work, for the general case and these three important election systems, clarifies the landscape of the 44 standard control types, for each pair collapsing or separating them, and also providing finergrained information on the separations.more » « less

Carleton, B. ; Chavrimootoo, Michael C. ; Hemaspaandra, Lane A. ; Narvaez, D. ; Taliancich, C. ; Welles, H. ( , Proceedings of the 22nd International Conference on Autonomous Agents and Multiagent Systems)Hemaspaandra et al. [6] and Carleton et al. [3, 4] found that many pairs of electoral (decision) problems about the same election sys tem coincide as sets (i.e., they are collapsing pairs), which had pre viously gone undetected in the literature. While both members of a collapsing pair certainly have the same decision complexity, there is no guarantee that the associated search problems also have the same complexity. For practical purposes, search problems are more relevant than decision problems. Our work focuses on exploring the relationships between the search versions of collapsing pairs. We do so by giving a framework that relates the complexity of search problems via efficient reduc tions that transform a solution from one problem to a solution of the other problem on the same input. We not only establish that the known decision collapses carry over to the search model, but also refine our results by determining for the concrete systems plurality, veto, and approval whether collapsing searchproblem pairs are polynomialtime computable or NPhard.more » « less