Hard enumeration problems in geometry and combinatorics
SIAM Journal on Algebraic and Discrete Methods
NP is as easy as detecting unique solutions
Theoretical Computer Science
The design and analysis of algorithms
The design and analysis of algorithms
Theoretical Computer Science
Complexity of generalized satisfiability counting problems
Information and Computation
The Complexity of Planar Counting Problems
SIAM Journal on Computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Local Problems, Planar Local Problems and Linear Time
CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
Satisfiability Parsimoniously Reduces to the Tantrix™ Rotation Puzzle Problem
Fundamenta Informaticae - Machines, Computations and Universality, Part I
Satisfiability parsimoniously reduces to the Tantrix™ rotation puzzle problem
MCU'07 Proceedings of the 5th international conference on Machines, computations, and universality
Some observations on holographic algorithms
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Satisfiability Parsimoniously Reduces to the Tantrix™ Rotation Puzzle Problem
Fundamenta Informaticae - Machines, Computations and Universality, Part I
Hi-index | 0.00 |
We prove that the Satisfiability (resp. planar Satisfiability) problem is parsimoniously P-time reducible to the 3-Colorability (resp. Planar 3-Colorability) problem, that means that the exact number of solutions is preserved by the reduction, provided that 3-colorings are counted modulo their six trivial color permutations. In particular, the uniqueness of solutions is preserved, which implies that Unique 3-Colorability is exactly as hard as Unique Satisfiability in the general case as well as in the planar case. A consequence of our result is the DP-completeness of Unique 3-Colorability and Unique Planar 3-Colorability under random P-time reductions. It also gives a finer and unified proof of the #P-completeness of #3-Colorability that was first obtained by Linial for the general case, and later by Hunt et al. for the planar case. Previous authors' reductions were either weakly parsimonious with a multiplication of the numbers of solutions by an exponential factor, or involved #P-complete intermediate counting problems derived from trivial "yes"-decision problems.