On Czerwinski's "P≠NP{\rm P} \neq {\rm NP} relative to a P{\rm P}-complete oracle"

In this paper, we take a closer look at Czerwinski's "${\rm P}\neq{\rm NP}$ relative to a ${\rm P}$-complete oracle" [Cze23]. There are (uncountably) infinitely-many relativized worlds where ${\rm P}$ and ${\rm NP}$ differ, and it is well-known that for any ${\rm P}$-complete problem $A$, ${\rm P}^A \neq {\rm NP}^A \iff {\rm P}\neq {\rm NP}$. The paper defines two sets ${\rm D}_{\rm P}$ and ${\rm D}_{\rm NP}$ and builds the purported proof of their main theorem on the claim that an oracle Turing machine with ${\rm D}_{\rm NP}$ as its oracle and that accepts ${\rm D}_{\rm P}$ must make $\Theta(2^n)$ queries to the oracle. We invalidate the latter by proving that there is an oracle Turing machine with ${\rm D}_{\rm NP}$ as its oracle that accepts ${\rm D}_{\rm P}$ and yet only makes one query to the oracle. We thus conclude that Czerwinski's paper [Cze23] fails to establish that ${\rm P} \neq {\rm NP}$

Search versus Search for Collapsing Electoral Control Types

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.Comment: The metadata's abstract is abridged due to arXiv.org's abstract-length limit. The paper itself has the unabridged (i.e., full) abstrac

Separating and Collapsing Electoral Control Types

[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.Comment: The arXiv.org metadata abstract is an abridged version; please see the paper for the full abstrac

Defying Gravity: On the Complexity of the Hanano Puzzle

Liu and Yang recently proved the Hanano Puzzle to be ${\rm NP}$-$\leq_m^p$-hard. We prove it is in fact ${\rm PSPACE}$-$\leq_m^p$-complete. Our paper introduces the notion of a planar grid and establishes a relationship between planar grids and instances of the Nondeterministic Constraint Logic (${\rm NCL}$) problem (a known ${\rm PSPACE}$-$\leq_m^p$-complete problem) by using graph theoretic methods, and uses this connection to guide an indirect many-one reduction from the ${\rm NCL}$ problem to the Hanano Puzzle. The technique introduced is versatile and can be reapplied to other games with gravity.Comment: 20 pages, 10 figure