Satisfiability Parsimoniously Reduces to the Tantrix(TM) Rotation Puzzle Problem

TL;DR

This paper demonstrates that the Tantrix(TM) rotation puzzle problem is computationally as hard as the satisfiability problem, including its counting and unique solution variants, establishing its complexity within the boolean hierarchy.

Contribution

It provides a parsimonious reduction from satisfiability to the Tantrix(TM) rotation puzzle, preserving solution uniqueness and complexity class implications.

Findings

The reduction is parsimonious, preserving solution counts.

Unique Tantrix(TM) puzzle problem is DP-complete.

The problem is as hard as the satisfiability problem and its variants.

Abstract

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem is NP-complete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the counting version and the unique version of this problem. We prove that the satisfiability problem parsimoniously reduces to the Tantrix(TM) rotation puzzle problem. In particular, this reduction preserves the uniqueness of the solution, which implies that the unique Tantrix(TM) rotation puzzle problem is as hard as the unique satisfiability problem, and so is DP-complete under polynomial-time randomized reductions, where DP is the second level of the boolean hierarchy over NP.