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In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times.
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 .
It runs in polynomial time on inputs that are in SUBSET-SUM if and only if P = NP: // Algorithm that accepts the NP-complete language SUBSET-SUM. // // this is a polynomial-time algorithm if and only if P = NP. // // "Polynomial-time" means it returns "yes" in polynomial time when // the answer should be "yes", and runs forever when it is "no".
The program is solvable in polynomial time if the graph has all undirected or all directed edges. Variants include the rural postman problem. [3]: ND25, ND27 Clique cover problem [2] [3]: GT17 Clique problem [2] [3]: GT19 Complete coloring, a.k.a. achromatic number [3]: GT5 Cycle rank; Degree-constrained spanning tree [3]: ND1
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.
On the one hand, there are many problems that have a solution space just as large, but can be solved in polynomial time (for example minimum spanning tree). On the other hand, there are NP-problems with at most one solution that are NP-hard under randomized polynomial-time reduction (see Valiant–Vazirani theorem).
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 decision problem is NEXPTIME-complete if it is in NEXPTIME, and every problem in NEXPTIME has a polynomial-time many-one reduction to it. In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer. Problems that are NEXPTIME-complete might be thought of as the hardest ...