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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
An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259 [13] 20 examples of periodic solutions to the three-body problem. In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of the same family of solutions: the Broucke–Hénon–Hadjidemetriou family. In this ...
"The problem of deciding whether the definite contour multiple integral of an elementary meromorphic function is zero over an everywhere real analytic manifold on which it is analytic", a consequence of the MRDP theorem resolving Hilbert's tenth problem. [5] Determining the domain of a solution to an ordinary differential equation of the form
Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the ...
Moravec's paradox: Logical thought is hard for humans and easy for computers, but picking a screw from a box of screws is an unsolved problem. Movement paradox: In transformational linguistics, there are pairs of sentences in which the sentence without movement is ungrammatical while the sentence with movement is not.
For example, the problem of factoring "Given a positive integer n, find a nontrivial prime factor of n." is a computational problem that has a solution, as there are many known integer factorization algorithms. A computational problem can be viewed as a set of instances or cases together with a, possibly empty, set of solutions for every ...
The most common problem being solved is the 0-1 knapsack problem, which restricts the number of copies of each kind of item to zero or one. Given a set of items numbered from 1 up to , each with a weight and a value , along with a maximum weight capacity ,
The opposite has also been claimed, for example by Karl Popper, who held that such problems do exist, that they are solvable, and that he had actually found definite solutions to some of them. David Chalmers divides inquiry into philosophical progress in meta-philosophy into three questions.