NP-Hard and k-EXPSPACE-Hard Cast Puzzles

Abstract

SUMMARYA disentanglement puzzle consists of mechanically interlinked pieces, and the puzzle is solved by disentangling one piece from another set of pieces. A cast puzzle is a type of disentanglement puzzle, where each piece is a zinc die-casting alloy. In this paper, we consider the generalized cast puzzle problem whose input is the layout of a finite number of pieces (polyhedrons) in the 3-dimensional Euclidean space. For every integer k ≥ 0, we present a polynomial-time transformation from an arbitrary k-exponential-space Turing machine M and its input x to a cast puzzle c 1 of size k-exponential in |x| such that M accepts x if and only if c 1 is solvable. Here, the layout of c 1 is encoded as a string of length polynomial (even if c 1 has size k-exponential). Therefore, the cast puzzle problem of size k-exponential is k-EXPSPACE-hard for every integer k ≥ 0. We also present a polynomial-time transformation from an arbitrary instance f of the SAT problem to a cast puzzle c 2 such that f is satisfiable if and only if c 2 is solvable.

A Polynomial-Time Reduction from the 3SAT Problem to the Generalized String Puzzle Problem

“…Chess [14] and Go [15] are known to be EXPTIME-hard. The current authors constructed a cast puzzle (a disentanglement puzzle consisting of zinc die-casting alloys) whose complexity is k-EXPSPACE-hard [16]. More information on games and puzzles can be found in [17].…”