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The non-constructive part here is the Robertson–Seymour theorem. Although it guarantees that there is a finite number of minor-minimal elements it does not tell us what these elements are. Therefore, we cannot really execute the "algorithm" mentioned above. But, we do know that an algorithm exists and that its runtime is polynomial.
A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive, showing an explicit bounding polynomial and algorithmic details, if the polynomial is not very low-order the algorithm might not be sufficiently efficient in practice.
Redwood City, California: Benjamin/Cummings Publishing Company, Inc. Appendix C includes impossibility of algorithms deciding if a grammar contains ambiguities, and impossibility of verifying program correctness by an algorithm as example of Halting Problem. Halava, Vesa (1997).
This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof .
To do this, instead of computing the conditional probability of failure, the algorithm computes the conditional expectation of Q and proceeds accordingly: at each interior node, there is some child whose conditional expectation is at most (at least) the node's conditional expectation; the algorithm moves from the current node to such a child ...
The algorithm then enters the main loop which is executed until all events in are avoided, at which point the algorithm returns the current assignment. At each iteration of the main loop, the algorithm picks an arbitrary satisfied event A (either randomly or deterministically) and resamples all the random variables that determine A .
An algorithm that verifies whether a given subset has sum zero is a verifier. Clearly, summing the integers of a subset can be done in polynomial time, and the subset sum problem is therefore in NP. The above example can be generalized for any decision problem.
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.