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Similarly, a problem hard for a class C is called C-hard, e.g. NP-hard. Normally, it is assumed that the reduction in question does not have higher computational complexity than the class itself. Therefore, it may be said that if a C-complete problem has a "computationally easy" solution, then all problems in "C" have an "easy" solution.
NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine. [2]
Although a solution to an NP-complete problem can be verified "quickly", there is no known way to find a solution quickly. That is, the time required to solve the problem using any currently known algorithm increases rapidly as the size of the problem grows.
This "yes-no" format is often equivalently stated as "accept-reject"; that is, an algorithm "accepts" an input string if the answer to the decision problem is "yes" and "rejects" if the answer is "no". While some problems cannot easily be expressed as decision problems, they nonetheless encompass a broad range of computational problems. [2]
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Get ready for all of today's NYT 'Connections’ hints and answers for #549 on Wednesday, December 11, 2024. Today's NYT Connections puzzle for Wednesday, December 11, 2024The New York Times.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
A singular solution in this stronger sense is often given as tangent to every solution from a family of solutions. By tangent we mean that there is a point x where y s (x) = y c (x) and y' s (x) = y' c (x) where y c is a solution in a family of solutions parameterized by c. This means that the singular solution is the envelope of the family of ...