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Concept
NP-hardness
Summary
In computational complexity theory, NP-hardness (non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem.
A more precise specification is: a problem H is NP-hard when every problem L in NP can be reduced in polynomial time to H; that is, assuming a solution for H takes 1 unit time, Hs solution can be used to solve L in polynomial time. As a consequence, finding a polynomial time algorithm to solve any NP-hard problem would give polynomial time algorithms for all the problems in NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists.
It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven. Moreover, the class P, in which all problems can be solved in polynomial time, is contained in the NP class.
A decision problem H is NP-hard when for every problem L in NP, there is a polynomial-time many-one reduction from L to H.
An equivalent definition is to require that every problem L in NP can be solved in polynomial time by an oracle machine with an oracle for H. Informally, an algorithm can be thought of that calls such an oracle machine as a subroutine for solving H and solves L in polynomial time if the subroutine call takes only one step to compute.
Another definition is to require that there be a polynomial-time reduction from an NP-complete problem G to H. As any problem L in NP reduces in polynomial time to G, L reduces in turn to H in polynomial time so this new definition implies the previous one. It does not restrict the class NP-hard to decision problems, and it also includes search problems or optimization problems.
If P ≠ NP, then NP-hard problems could not be solved in polynomial time.
Some NP-hard optimization problems can be polynomial-time approximated up to some constant approximation ratio (in particular, those in APX) or even up to any approximation ratio (those in PTAS or FPTAS).
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