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Class of decision problems which contains the hardest problems in NP. Each NP-complete problem has to be in NP. NP-easy At most as hard as NP, but not necessarily in NP. NP-equivalent Decision problems that are both NP-hard and NP-easy, but not necessarily in NP. NP-intermediate If P and NP are different, then there exist decision problems in ...
Quadratic programming (NP-hard in some cases, P if convex) Subset sum problem [3]: SP13 Variations on the Traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric.
Informally, an NP-complete problem is an NP problem that is at least as "tough" as any other problem in NP. NP-hard problems are those at least as hard as NP problems; i.e., all NP problems can be reduced (in polynomial time) to them. NP-hard problems need not be in NP; i.e., they need not have solutions verifiable in polynomial time.
Euler diagram for P, NP, NP-complete, and NP-hard set of problems (excluding the empty language and its complement, which belong to P but are not NP-complete) Main article: P versus NP problem The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time ), an algorithm ...
In computational complexity, an NP-complete (or NP-hard) problem is weakly NP-complete (or weakly NP-hard) if there is an algorithm for the problem whose running time is polynomial in the dimension of the problem and the magnitudes of the data involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes.
"NP-complete problems are the most difficult known problems." Since NP-complete problems are in NP, their running time is at most exponential. However, some problems have been proven to require more time, for example Presburger arithmetic. Of some problems, it has even been proven that they can never be solved at all, for example the halting ...