[HHM20] discovered, for 7 pairs (C,D) of seemingly distinct standard electoral control types, that C and D are identical: For each input I and each election system, I is a Yes instance of both C and D, or of neither. Surprisingly this had gone undetected, even as the field was score-carding how many std. 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 std. 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 7. We for 3 central election systems determine which control pairs are identical ("collapse") with respect to those systems, and we explore containment/incomparability relationships between control pairs. For approval voting, which has a different "type" for its votes, [HHM20]'s 7 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 1 additional collapse for veto and none for plurality. We prove that each of the 3 election systems mentioned have no collapses other than those inherited from [HHM20] or added here. But we show many new containment relationships that hold between some separating control pairs, and for each separating pair of std. 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 3 important election systems, clarifies the landscape of the 44 std. control types, for each pair collapsing or separating them, and also providing finer-grained information on the separations.

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.

Backdoors and backbones of Boolean formulas are hidden structural properties. A natural goal, already in part realized, is that solver algorithms seek to obtain substantially better performance by exploiting these structures. However, the present paper is not intended to improve the performance of SAT solvers, but rather is a cautionary paper. In particular, the theme of this paper is that there is a potential chasm between the existence of such structures in the Boolean formula and being able to effectively exploit them. This does not mean that these structures are not useful to solvers. It does mean that one must be very careful not to assume that it is computationally easy to go from the existence of a structure to being able to get one's hands on it and/or being able to exploit the structure. For example, in this paper we show that, under the assumption that P $\neq$ NP, there are easily recognizable families of Boolean formulas with strong backdoors that are easy to find, yet for which it is hard (in fact, NP-complete) to determine whether the formulas are satisfiable. We also show that, also under the assumption P $\neq$ NP, there are easily recognizable sets of Boolean formulas for which it is hard (in fact, NP-complete) to determine whether they have a large backbone.

We discuss the connection between computational social choice (comsoc) and computational complexity. We stress the work so far on, and urge continued focus on, two less-recognized aspects of this connection. Firstly, this is very much a two-way street: Everyone knows complexity classification is used in comsoc, but we also highlight benefits to complexity that have arisen from its use in comsoc. Secondly, more subtle, less-known complexity tools often can be very productively used in comsoc.

A backbone of a boolean formula F is a collection S of its variables for which there is a unique partial assignment a S such that F [ a S ] is satisfiable (Monasson et al. 1999; Williams, Gomes, and Selman 2003).  This paper studies the nontransparency of backbones.  We show that, under the widely believed assumption that integer factoring is hard, there exist sets of boolean formulas that have obvious, nontrivial backbones yet finding the values, a S , of those backbones is intractable.  We also show that, under the same assumption, there exist sets of boolean formulas that obviously have large backbones yet producing such a backbone S is intractable.  Further, we show that if integer factoring is not merely worst-case hard but is frequently hard, as is widely believed, then the frequency of hardness in our two results is not too much less than that frequency.

Scoring systems are an extremely important class of election systems. A length-m (so-called) scoring vector applies only to m-candidate elections. To handle general elections, one must use a family of vectors, one per length. The most elegant approach to making sure such families are "family-like'' is the recently introduced notion of (polynomial-time uniform) pure scoring rules, where each scoring vector is obtained from its precursor by adding one new coefficient. We obtain the first dichotomy theorem for pure scoring rules for a control problem. In particular, for constructive control by adding voters (CCAV), we show that CCAV is solvable in polynomial time for k-approval with k<=3, k-veto with k<=2, every pure scoring rule in which only the two top-rated candidates gain nonzero scores, and a particular rule that is a "hybrid" of 1-approval and 1-veto. For all other pure scoring rules, CCAV is NP-complete. We also investigate the descriptive richness of different models for defining pure scoring rules, proving how more rule-generation time gives more rules, proving that rationals give more rules than do the natural numbers, and proving that some restrictions previously thought to be "w.l.o.g." in fact do lose generality.

For many election systems, bribery (and related) attacks have been shown NP-hard using constructions on combinatorially rich structures such as partitions and covers. It is important to learn how robust these hardness protection results are, in order to find whether they can be relied on in practice. This paper shows that for voters who follow the most central political-science model of electorates — single-peaked preferences — those protections vanish. By using single-peaked preferences to simplify combinatorial covering challenges, we show that NP-hard bribery problems — including those for Kemeny and Llull elections- — fall to polynomial time. By using single-peaked preferences to simplify combinatorial partition challenges, we show that NP-hard partition-of-voters problems fall to polynomial time. We furthermore show that for single-peaked electorates, the winner problems for Dodgson and Kemeny elections, though Θ 2 p -complete in the general case, fall to polynomial time. And we completely classify the complexity of weighted coalition manipulation for scoring protocols in single-peaked electorates.