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

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.