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This means that they are at least as hard as any problem in the class . If a problem is C {\displaystyle C} -hard (with respect to polynomial time reductions), then it cannot be solved by a polynomial-time algorithm unless the computational hardness assumption P ≠ C {\displaystyle P\neq C} is false.
A problem is hard for a class of problems if every problem in can be reduced to . Thus no problem in C {\displaystyle C} is harder than X {\displaystyle X} , since an algorithm for X {\displaystyle X} allows us to solve any problem in C {\displaystyle C} .
A problem is hard for a class of problems C if every problem in C can be polynomial-time reduced to . Thus no problem in C is harder than X {\displaystyle X} , since an algorithm for X {\displaystyle X} allows us to solve any problem in C with at most polynomial slowdown.
The Subgraph Isomorphism problem is NP-complete. The graph isomorphism problem is suspected to be neither in P nor NP-complete, though it is in NP. This is an example of a problem that is thought to be hard, but is not thought to be NP-complete. This class is called NP-Intermediate problems and exists if and only if P≠NP.
NP-hard Class of problems which are at least as hard as the hardest problems in NP. Problems that are NP-hard do not have to be elements of NP; indeed, they may not even be decidable. NP-complete Class of decision problems which contains the hardest problems in NP. Each NP-complete problem has to be in NP. NP-easy
In computational complexity theory, co-NP is a complexity class.A decision problem X is a member of co-NP if and only if its complement X is in the complexity class NP.The class can be defined as follows: a decision problem is in co-NP if and only if for every no-instance we have a polynomial-length "certificate" and there is a polynomial-time algorithm that can be used to verify any purported ...
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To find a maximum clique, one can systematically inspect all subsets, but this sort of brute-force search is too time-consuming to be practical for networks comprising more than a few dozen vertices. Although no polynomial time algorithm is known for this problem, more efficient algorithms than the brute-force search are known.