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Using the notion of visibility representations, our paper establishes a new property of in stances of the Nondeterministic Constraint Logic (NCL) problem (a PSPACEcomplete problem that is very convenient to prove the PSPACEhardness of reversible games with pushing blocks). Direct use of this property introduces an explosion in the number of gadgets needed to show PSPACEhardness, but we show how to bring that number from 32 down to only three in general, and down to two in a specific case! We propose it as a step towards a broader and more general framework for studying games with irreversible gravity, and use this connection to guide an indirect polynomialtime manyone reduction from the NCL problem to the Hanano Puzzle—which is NPhard—to prove it is in fact PSPACEcomplete.more » « less

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

Following the seminal work of Bartholdi et al. [2], there has been a slew of research on the complexity of constructive and destructive control for specific election systems (a.k.a. voting rules), which was driven by the field’s desire to find a natural election system that is “resistant” to as many control attacks (types) as possible. While this race was happening, many proofs were devised for a variety of election systems, and yet unbeknownst to many, several control attacks were in fact exactly the same (when viewed as decision problems, which is the common framework). Hemaspaandra et al. [14] were the first to make this observation, demonstrating that there was a general lack of understanding of the standard control attacks. My work continues this line of research in three ways: (1) determining the relationships of electoral control types both in the “general” setting and in concrete settings, (2) finding axiomatic sufficient conditions to determine if a particular equality between control types (a.k.a. collapse) occurs, and (3) linking results in the more abstract decision model to the more explicit search model.more » « less

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

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.more » « less

Cai and Hemachandra used iterative constantsetting 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 easytofind “targets” used in iterative constantsetting. In particular, we show that even less restrictive gapsize upper bounds regarding the targets allow one to capture ambiguitylimited classes. Through a flexible, metatheorembased approach, we do so for a wide range of classes including the logarithmicambiguity version of Valiant’s unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, Pprintable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstantambiguity analogues of UP. As an application of our work, we prove that the Lenstra–Pomerance–Wagstaff Conjecture implies that all O(log log n)ambiguity NP sets are in the restricted counting class RC_PRIMES .more » « less