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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 ...
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.
In this theory, the class P consists of all decision problems (defined below) solvable on a deterministic sequential machine in a duration polynomial in the size of the input; the class NP consists of all decision problems whose positive solutions are verifiable in polynomial time given the right information, or equivalently, whose solution can ...
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 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 ...
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.
"Harder" means having a higher estimate of the required computational resources in a given context (e.g., higher time complexity, greater memory requirement, expensive need for extra hardware processor cores for a parallel solution compared to a single-threaded solution, etc.). The existence of a reduction from A to B can be written in the ...