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In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer.
Many limits derived in terms of physical constants and abstract models of computation in computer science are loose. [12] Very few known limits directly obstruct leading-edge technologies, but many engineering obstacles currently cannot be explained by closed-form limits.
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 probably not NP-hard (unless ...
Hence, the Small Set Expansion Hypothesis, which postulates that SSE is hard to approximate, is a stronger (but closely related) assumption than the Unique Game Conjecture. [25] Some approximation problems are known to be SSE-hard [26] (i.e. at least as hard as approximating SSE).
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. [1] Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements.
For example, one of the long-standing open questions in computer science is to determine whether there is an algorithm that outperforms the 2-approximation for the Steiner Forest problem by Agrawal et al. [3] The desire to understand hard optimization problems from the perspective of approximability is motivated by the discovery of surprising ...
Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.
On the other hand, a problem is AI-Hard if and only if there is an AI-Complete problem that is polynomial time Turing-reducible to . This also gives as a consequence the existence of AI-Easy problems, that are solvable in polynomial time by a deterministic Turing machine with an oracle for some problem.