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Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits. [14] There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests.
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. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [3]: ND22, ND23
A solution to these problems is offered by segmented sieves, where only portions of the range are sieved at a time. [10] These have been known since the 1970s, and work as follows: [9] [11] Divide the range 2 through n into segments of some size Δ ≤ √ n. Find the primes in the first (i.e. the lowest) segment, using the regular sieve.
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 the region of NP that fall between P and the NP-complete problems.
P is known to contain many natural problems, including the decision versions of linear programming, and finding a maximum matching.In 2002, it was shown that the problem of determining if a number is prime is in P. [1] The related class of function problems is FP.
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In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices ; the length of a path may either be measured by its number of edges, or (in weighted graphs ) by the sum of the weights of its ...