Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.05518

Sorting by pile shuffles on queue-like and stack-like piles can be hard

Kyle B. Treleaven
View original source ↗Read source document ↗Export metadata ↓

Abstract

Inspired by a common technique for shuffling a deck of cards on a table without riffling, we continue the study of a prequel paper on the pile shuffle and its capabilities as a sorting device. We study two sort feasibility problems of general interest concerning pile shuffle, first introduced in the prequel. These problems are characterized by: (1) bounds on the number of sequential rounds of shuffle, and piles created in each round; (2) the use of a heterogeneous mixture of queue-like and stack-like piles, as when each round of shuffle may have a combination of face-up and face-down piles; and (3) the ability of the dealer to choose the types of piles used during each round of shuffle. We prove by a sequence of reductions from the Boolean satisfiability problem (SAT) that the more general problem is NP-Hard. We leave as an open question the complexity of its arguably more natural companion, but discuss avenues for further investigation. Our analysis leverages a novel framework, introduced herein, which equates instances of shuffle to members of a particular class of deterministic finite automata.

Explore related subjects

cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kyle B. Treleaven. 2025-06-05. Sorting by pile shuffles on queue-like and stack-like piles can be hard. https://arxiv.org/abs/2506.05518

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Convex Networks Remain Hard to Certify: Dimension-Accuracy Barriers for Lipschitz Constants

Input-convex neural networks permit globally tractable minimization over their inputs, so one might expect their global regularity to be tractable in low input dimension. We prove exact and accuracy-sensitive barriers to this expectation. Given a bias-free one-hidden-layer ReLU network $f(x)=\sum_{r=1}^n \mathrm{ReLU}(a_r^\top x)$ with unit positive output weights, deciding whether its global Euclidean Lipschitz constant is at least a rational threshold is NP-complete and W[1]-hard when parameterized by the input dimension $d$. The same holds on the unit ball and with integral first-layer weights having at most nine nonzeros. More sharply, no deterministic multiplicative approximation scheme runs in $g(d)\mathrm{poly}(\mathcal B,1/\varepsilon)$ time unless FPT equals W[1]. Under the Exponential Time Hypothesis, no such algorithm runs in $g(d)(\mathcal B+1/\varepsilon)^{o(d/\log d)}$ time. Thus accuracy cannot have a polynomial dependence separated from dimension. The exact result resolves the Euclidean case of an open problem posed at COLT 2025 and left open by the ICLR 2026 parameterized hardness theory for general two-layer networks. The approximation barrier is specific to generator-presented zonotopes, complementing known $(1/\varepsilon)^{O(d)}$-time schemes and an analogous barrier for halfspace-presented polytopes. Our lifted-selector reduction has an inverse-polynomial radial gap, proved through a quantitative theorem for rational cyclic zonogons. Equivalently, the results apply to Euclidean zonotope radius and positive-semidefinite binary quadratic maximization parameterized by rank. Convexity makes minimization easy, but it does not make global sensitivity fixed-parameter tractable or permit a dimension-separated fully polynomial accuracy guarantee.

cs.CC↗
arXiv2026

Lettericity Is NP-Complete

The lettericity of a graph $G$ is the smallest size of a set $Σ$ such that there exist $w_1, \ldots, w_{|V(G)|} \in Σ$ and a decoder $D \subseteq Σ^2$ for which $G$ is isomorphic to the letter graph $(\{1, \ldots, |V(G)|\}, \{ij : 1 \le i < j \le |V(G)|, w_iw_j \in D\})$. It took around two decades of the study of lettericity for, in the simpler case of paths, a closed-form expression for its lettericity to be derived; this suggests that the question of whether the lettericity of an arbitrary graph can be computed in polynomial time is nontrivial. Indeed, this question has been raised repeatedly as an open problem in recent literature. We solve this problem by showing that the lettericity problem on arbitrary graphs is NP-complete (Theorem 10). We also prove that the coloring extension problem (the same problem as lettericity, with the added condition that if $f$ is the isomorphism mapping from $G$ to the letter graph, $w_{f(v)} = χ(v)$ for a given coloring $χ$ of $G$) is NP-complete (Theorem 12). We also resolve the open problem of classifying the complexity of the word extension problem, which is the same problem as lettericity except that the $w_i$ are fixed; we show it to be NP-complete (Theorem 13), which, in tandem with our NP-completeness result for coloring extension, contrasts with the known result that when the constraint of the coloring extension problem and the constraint of the word extension problem are both applied to lettericity, lettericity can be decided in polynomial time. Additionally, we use the reduction in the NP-completeness proof to show that unless the Exponential Time Hypothesis is false, there cannot exist a deterministic algorithm to decide whether the lettericity of an $n$-vertex graph is at most $k$ in time $2^{o(n)}$, even when $n = 6k$ (Theorem 11).

cs.CC↗
arXiv2026

NP-Hardness of Bounded Distance Decoding for Reed-Solomon Codes

For an $[n,K]$ Reed--Solomon code, the covering radius is $n-K$. Gandikota, Ghazi, and Grigorescu proved deterministic NP-hardness of bounded-distance decoding when the decoding radius is $d$ below the covering radius for every $1\le d\le c\log n/\log\log n$, where $c>0$ is an absolute constant. We prove that, for every fixed rational $0<α<1/2$, bounded-distance decoding is NP-complete under deterministic polynomial-time many-one reductions over explicitly represented finite extension fields for the additive gap $d=\lfloor n^α\rfloor$ below the covering radius. The hard codes have odd block length~$n$, dimension $K=(n+1)/2-d$, decoding radius $(n-1)/2$, and rate tending to $1/2$. The alphabet size is subexponential in the evaluation set size: for a fixed $0<η<1$ depending only on $α$, it is $2^{Θ(n^η\log n)}=2^{o(n)}$. The proof passes through moments subset sum on $n-1$ nonzero field elements, with required subset size $(n-1)/2$ and $d$ prescribed moments. The arithmetic ingredient is a uniform positive-completion theorem over prime fields $\mathbb{F}_q$ with $q\ge d^{2+ρ}$, for any fixed $ρ>0$. A sharper form follows from a higher-dimensional point-count estimate based on Deligne's theorem; the weaker form used in our reduction is proved more elementarily using additive-character orthogonality, the one-variable Weil bound, a moment identity of order $2d$, and Newton identities. A universal completion pool, an extension-field quotient construction, and a deterministic linear-size simultaneous power condenser complete the reduction.

cs.CC↗