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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.
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 .
Remains NP-complete for 3-sets. Solvable in polynomial time for 2-sets (this is a matching). [2] [3]: SP2 Finding the global minimum solution of a Hartree-Fock problem [37] Upward planarity testing [8] Hospitals-and-residents problem with couples; Knot genus [38]
In computational complexity theory, a computational problem H is called NP-hard if, for every problem L which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from L to H. That is, assuming a solution for H takes 1 unit time, H ' s solution can be used to solve L in polynomial time.
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 ...
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.
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.
Karmarkar's algorithm falls within the class of interior-point methods: the current guess for the solution does not follow the boundary of the feasible set as in the simplex method, but moves through the interior of the feasible region, improving the approximation of the optimal solution by a definite fraction with every iteration and ...