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. 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.

Cai and Hemachandra used iterative constant-setting to prove that Few⊆⊕P (and thus that Fewp⊆⊕P). In this paper, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed “nongappy”-ness) of the easy-to-find “targets” used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant’s unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra–Pomerance–Wagstaff Conjecture implies that all \((\mathcal {O}(1) + \log \log n)\) -ambiguity NP sets are in the restricted counting class RCPRIMES.

Anna and Belle, who now have become so aware of their role as guides in this book that they can even refer to the book's content, meet on what will be a very exciting day for them. Let's listen in on their conversation.

Electoral control refers to attacking elections by adding, deleting, or partitioning voters or candidates. Hemaspaandra, Hemaspaandra, and Menton recently discovered, for seven pairs (T, T′) of seemingly distinct standard electoral control types, that T and T′ are in practice 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 previously gone undetected even as the field was score-carding how many standard control types various 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 perhaps other pairs of control types are identical, and so work still is being needlessly duplicated. We completely determine, for all standard control types, which pairs are, for elections whose votes are linear orderings of the candidates, always identical. In particular, we prove that no identical control pairs exist beyond the known seven. We also for three central election systems completely determine which control pairs are identical (“collapse”) with respect to those particular election systems, and we also explore containment and incomparability relationships between control pairs. For approval voting, which has a different “type” for its votes, Hemaspaandra, Hemaspaandra, and Menton’s seven collapses still hold (since we observe that their argument applies to all election systems). However, we find 14 additional collapses that hold for approval voting but do not hold for some election systems whose votes are linear orderings of the candidates. We find one new collapse for veto elections and none for plurality. We prove that each of the three election systems mentioned have no collapses other than those inherited from Hemaspaandra, Hemaspaandra, and Menton or added in the present paper. We establish many new containment relationships between separating control pairs, and for each separating pair of standard control types classify its separation in terms of either 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 finer-grained information on the separations.

The penultimate complexity column during my time as complexity column editor is Parting Thoughts and Parting Shots (Read On for Details on How to Win Valuable Prizes!), by Eric Allender. Warmest thanks to Eric for sharing with the field this wide range of (potentially lucrative!) challenges.

It has been a tremendous treat being the SIGACT News Complexity Theory Column editor for these past thirty years. I thought about what to say here, and realized it is pretty simple: Thank you.

My deepest thanks to Nutan Limaye, Srikanth Srinivasan, and Sebastien Tavenas for their fascinating article, Lower bounds Against Constant-Depth Algebraic Circuits.

As I write this (in January), it is -7F (and -17F with the wind chill factor) here in Western New York, and looking at the pictures of this guest article's authors, I'm thinking that (from the outdoor pictures) Sweden and Switzerland are looking like pretty toasty places! (Robert's picture seems to be taken inside, but I see that currently the temperature is -15F in Montreal, so one can't blame him: brrrrrr! For completeness, Lund is currently 34F and Lausanne is currently 38F, but the time-zone di erence is helping them a bit... and also, for all I know those might be summer snaps.)

Juris Hartmanis was so preternaturally wise and brilliant that in time people may find it hard to believe that someone so extraordinary existed in this world. Yet as I write this, three months to the day after Juris passed away, it still seems impossible that such a force of nature can no longer be part of this world.

Prior work on the complexity of bribery assumes that the bribery happens simultaneously, and that the briber has full knowledge of all voters' votes. But neither of those assumptions always holds. In many real-world settings, votes come in sequentially, and the briber may have a use-it-or-lose-it moment to decide whether to bribe/alter a given vote, and at the time of making that decision, the briber may not know what votes remaining voters are planning on casting. In this paper, we introduce a model for, and initiate the study of, bribery in such an online, sequential setting. We show that even for election systems whose winner-determination problem is polynomial-time computable, an online, sequential setting may vastly increase the complexity of bribery, in fact jumping the problem up to completeness for high levels of the polynomial hierarchy or even PSPACE. On the other hand, we show that for some natural, important election systems, such a dramatic complexity increase does not occur, and we pinpoint the complexity of their bribery problems in the online, sequential setting.

This issue's column (which comes immediately after some memories of and comments on Juris Hartmanis, who passed away a few days ago), by Eleni Bakali, Aggeliki Chalki, Andreas Göbel, Aris Pagourtzis, and Stathis Zachos, is a tour--- A Panorama of Counting Problems the Decision Version of which is in P ---of the exciting world of counting functions whose decision version (the set of inputs on which the function evaluates to zero) is in P.

Juris Hartmanis was so preternaturally wise and brilliant that in time people may find it hard to believe that someone so extraordinary existed in this world. Yet as I write this, three months to the day after Juris passed away, it still seems impossible that such a force of nature can no longer be part of this world.

This issue's complexity theory column is by Ben Lee Volk on algebraic natural proofs. My warmest thanks to Ben Lee for his terrific article.

My deepest thanks to Beatrice and Carlo for their fascinating article, Quantum Finite Automata: From Theory to Practice. Regarding the extent to which their article brings to life both parts of its subtitle... wow! And I think Section 4 is an absolute first for this column; please don't miss it!

Warmest thanks to Rafael Pass and Muthu Venkitasubramaniam for this issue's guest column, "Average-Case Complexity Through the Lens of Interactive Puzzles." When I mentioned to them that my introduction would have a section on Alan Selman's passing, they immediately wrote back that they were very sorry to hear of Alan's passing, and mentioned (as you will see discussed in the second page of their article), "The main problem that we are addressing actually goes back to a paper of Even, Selman, and Yacobi from 1984: "The Complexity of Promise Problems with Applications to Public-Key Cryptography'." It is beautiful, and a tribute to the lasting influence of Alan's research, that in the 2020s his work from many decades earlier is helping shape the field's dialogue.

This paper uses structural complexity theory to study whether there is a chasm between knowing an object exists and getting one's hands on the object or its properties. In particular, we study 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 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 is intractable. Furthermore, 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. These results hold even if one's assumptions are, respectively, P≠NP∩coNP or that some NP∩coNP problem is frequently hard.

Warmest thanks to Alexander Knop, Shachar Lovett, Sam McGuire, and Weiqiang Yuan for this issue's guest column, \Models of computation between decision trees and communication." (Their article came in early, and I wrote back thanking them for being three days early, mentioning that that happens surprisingly rarely. Then two days later, I said, \Wow!", as I realized that they actually had somehow|even during a remote-teaching, social-distancing time period|prepared and sent in their article a month and three days early.)

My deepest thanks to Sabine Broda, Antonio Machiavelo, Nelma Moreira, and Rogerio Reis for welcoming the year 2020 with their exciting tutorial, \Analytic Combinatorics and Descriptional Complexity of Regular Languages on Average."

Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If integer factoring is hard, then for election manipulation, election bribery, and some types of election control, there are election systems for which recognizing which instances can be successfully manipulated is in polynomial time but producing the successful manipulations cannot be done in polynomial time.