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A polynomial-time many-one reduction from a problem A to a problem B (both of which are usually required to be decision problems) is a polynomial-time algorithm for transforming inputs to problem A into inputs to problem B, such that the transformed problem has the same output as the original problem.
As described in the example above, there are two main types of reductions used in computational complexity, the many-one reduction and the Turing reduction.Many-one reductions map instances of one problem to instances of another; Turing reductions compute the solution to one problem, assuming the other problem is easy to solve.
A polynomial-time counting reduction is usually used to transform instances of a known-hard problem into instances of another problem that is to be proven hard. It consists of two functions f {\displaystyle f} and g {\displaystyle g} , both of which must be computable in polynomial time .
An approximation-preserving reduction is a pair of functions (,) (which often must be computable in polynomial time), such that: f maps an instance x of A to an instance ′ of B. g maps a solution ′ of B to a solution y of A.
In computational complexity theory, a PTAS reduction is an approximation-preserving reduction that is often used to perform reductions between solutions to optimization problems. It preserves the property that a problem has a polynomial time approximation scheme (PTAS) and is used to define completeness for certain classes of optimization ...
If one rounds off some of the least significant digits of the profit values then they will be bounded by a polynomial and 1/ε where ε is a bound on the correctness of the solution. This restriction then means that an algorithm can find a solution in polynomial time that is correct within a factor of (1-ε) of the optimal solution. [26]
In this case, the proof shows that a solution of Sudoku in polynomial time could also be used to complete Latin squares in polynomial time. [12] This in turn gives a solution to the problem of partitioning tri-partite graphs into triangles, [ 13 ] which could then be used to find solutions for the special case of SAT known as 3-SAT, [ 14 ...
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...