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

A disentanglement puzzle consists of mechanically interlinked pieces, and the puzzle is solved by disentangling one piece from another set of pieces. A string puzzle consists of strings entangled with one or more wooden pieces. We consider the generalized string puzzle problem whose input is the layout of strings and a wooden board with holes embedded in the 3-dimensional Euclidean space. We present a polynomial-time transformation from an arbitrary instance？？ of the 3SAT problem to a string puzzle s such that？？ is satisfiable if and only if s is solvable. Therefore, the generalized string puzzle problem is NP-hard.