NP-hardness

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.^[1][2] As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists.^[3] It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven.^[4] A simple example of an NP-hard problem is the subset sum problem.

Informally, if H is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are provably not NP-hard (unless P=NP).^[5]