The Three-Color and Two-Color Tantrix(TM) Rotation Puzzle Problems are NP-Complete via Parsimonious Reductions

TL;DR

This paper proves that the three-color and two-color Tantrix rotation puzzle problems are NP-complete, extending the understanding of their computational complexity and answering open questions about their difficulty.

Contribution

It establishes NP-completeness for 3-TRP and 2-TRP, and shows their infinite variants are undecidable, providing new insights into these puzzle problems' complexity.

Findings

3-TRP and 2-TRP are NP-complete.

Unique-3-TRP and Unique-2-TRP are DP-complete.

Infinite variants of 3-TRP and 2-TRP are undecidable.

Abstract

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem with four colors is NP-complete, and they showed that the infinite variant of this problem is undecidable. In this paper, we study the three-color and two-color Tantrix(TM) rotation puzzle problems (3-TRP and 2-TRP) and their variants. Restricting the number of allowed colors to three (respectively, to two) reduces the set of available Tantrix(TM) tiles from 56 to 14 (respectively, to 8). We prove that 3-TRP and 2-TRP are NP-complete, which answers a question raised by Holzer and Holzer in the affirmative. Since our reductions are parsimonious, it follows that the problems Unique-3-TRP and Unique-2-TRP are DP-complete under randomized reductions. We also show that the another-solution problems associated with 4-TRP, 3-TRP, and 2-TRP are NP-complete. Finally, we prove that the infinite variants of 3-TRP and 2-TRP are undecidable.