P versus NP problem
Does P mean "easy"?

All of the above discussion has assumed that P means "easy" and "not in P" means "difficult", an assumption known as Cobham's thesis. It is a common assumption in complexity theory; but there are caveats.

First, it can be false in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents, rendering it impractical. For example, the problem of deciding whether a graph G contains H as a minor, where H is fixed, can be solved in a running time of O (n), where n is the number of vertices in G. However, the big O notation hides a constant that depends superexponentially on H. The constant is greater than 2 ↑ ↑ (2 ↑ ↑ (2 ↑ ↑ (h / 2))) {\displaystyle 2\uparrow \uparrow (2\uparrow \uparrow (2\uparrow \uparrow (h/2)))} (using Knuth's up-arrow notation), and where h is the number of vertices in H.

On the other hand, even if a problem is shown to be NP-complete, and even if P ≠ NP, there may still be effective approaches to the problem in practice. There are algorithms for many NP-complete problems, such as the knapsack problem, the traveling salesman problem, and the Boolean satisfiability problem, that can solve to optimality many real-world instances in reasonable time. The empirical average-case complexity (time vs. problem size) of such algorithms can be surprisingly low. An example is the simplex algorithm in linear programming, which works surprisingly well in practice; despite having exponential worst-case time complexity, it runs on par with the best known polynomial-time algorithms.

Finally, there are types of computations which do not conform to the Turing machine model on which P and NP are defined, such as quantum computation and randomized algorithms.
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