2 research outputs found
Satisfiability Parsimoniously Reduces to the Tantrix(TM) Rotation Puzzle Problem
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.Comment: 19 pages, 16 figures, appears in the Proceedings of "Machines,
Computations and Universality" (MCU 2007
The Three-Color and Two-Color Tantrix(TM) Rotation Puzzle Problems are NP-Complete via Parsimonious Reductions
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.Comment: 30 pages, 25 figure