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

On Unique Satisfiability and the Threshold Behavior of Randomized Reductions

The research presented in this paper is motivated by the some new results on the complexity of the unique satisfiability problem, USAT. These results, which are shown for the first time in this paper, are: ffl if USATj P m USAT, then D P = co-D P and PH collapses. ffl if USAT 2 co-D P , then PH collapses. ffl if USAT has OR! , then PH collapses. The proofs of these results use only the fact that USAT is complete for D P under randomized reductions---even though the probability bound of these reductions may be low. Furthermore, these results show that the structural complexity of USAT and of D P many-one complete sets are very similar, and so they lend support to the argument that even sets complete under "weak" randomized reductions can capture the properties of the many-one complete sets. However, under these "weak" randomized reductions, USAT is complete for P SAT[log n] as well, and in this case, USAT does not capture the properties of the sets many-one complete for ..