Parsimonious reduction

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In computational complexity theory and game complexity, a parsimonious reduction is a transformation from one problem to another (a reduction) that preserves the number of solutions.

More formally, parsimonious reductions are defined for problems in nondeterministic complexity classes such as NP, which are defined by a verifier algorithm whose input is a pair of strings (an instance of the problem and a candidate solution) and whose output is true when the candidate solution is a valid solution of the instance. For example, in the Boolean satisfiability problem, the instance might be a Boolean expression and the candidate solution might be a truth assignment to its variables; a valid truth assignment is one that makes the expression evaluate to true. A parsimonious reduction from one problem X of this type to another problem Y is an algorithmic transformation from instances of X to instances of Y such that the number of solutions of an instance of X equals the number of solutions of the transformed instance of Y.[1]

Just as many-one reductions are important for proving NP-completeness, parsimonious reductions are important for proving completeness for counting complexity classes such as ♯P.[1] Because parsimonious reductions preserve the property of having a unique solution, they are also used in game complexity, to show the hardness of puzzles such as sudoku where the uniqueness of the solution is an important part of the definition of the puzzle.[2]

Specific types of parsimonious reductions may be defined by the computational complexity or other properties of the transformation algorithm. For instance, a polynomial-time parsimonious reduction is one in which the transformation algorithm takes polynomial time. These are the types of reduction used to prove ♯P-completeness.[1] In parameterized complexity, fpt parsimonious reductions are used; these are parsimonious reductions whose transformation is a fixed-parameter tractable algorithm and that map bounded parameter values to bounded parameter values by a computable function.[3]

Polynomial-time parsimonious reductions are a special case of a more general class of reductions for counting problems, the polynomial-time counting reductions.[4]


  1. ^ a b c Goldreich, Oded (2008), Computational Complexity: A Conceptual Perspective, Cambridge University Press, pp. 203–204, ISBN 9781139472746
  2. ^ Yato, Takayuki; Seta, Takahiro (2003), "Complexity and Completeness of Finding Another Solution and Its Application to Puzzles", IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E86-A (5): 1052–1060
  3. ^ Flum, J.; Grohe, M. (2006), Parameterized Complexity Theory, EATCS Texts in Theoretical Computer Science, Springer, p. 363, ISBN 9783540299530
  4. ^ Gomes, Carla P.; Sabharwal, Ashish; Selman, Bart (2009), "Chapter 20. Model Counting", in Biere, Armin; Heule, Marijn; van Maaren, Hans; Walsh, Toby, Handbook of Satisfiability (PDF), Frontiers in Artificial Intelligence and Applications, 185, IOS Press, pp. 633–654, ISBN 9781586039295. See in particular pp. 634–635.