By Avella P., Boccia M., Sforza A.

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It is clear that this is a decision problem requires “yes” or “no” answer. Sorting a set of integers also is another example of P class problems. This is because, an algorithm can be found capable of solving the problem in polynomial time. A classic known algorithm for sorting is select method. 2 Class NP NP is the class of decision problems for which there exists a proof system such that the proofs are succinct and easy to check. 30 M. Avazbeigi In fact, in order to prove a problem is in NP, we do not require that there should exists an efficient way to find a proof of x when x 2 X; only there should exits an efficient way to check the validity of a proposed short proof (Brassard and Bratley 1988).

The distance between points inside a rectangle is taken to be zero. The distance between a point pi outside a rectangle R and the point inside that rectangle is taken to be the distance between pi and the closest point to it on the boundary of R which could either be a corner point of R, or the projection of pi onto R. note that since R is closed and convex, such a point indeed exists. 56 E.

Moradi and M. 3 Square Euclidean Distance with Point Facilities In some facility location problems, cost is not a simple linear function of distance. As an example, the cost associated with the response of a fire truck to a fire is expected to be nonlinear with distance. X / can take on a number of different formulations. X / treated in this chapter is the gravity problem. Suppose that cost is proportional to the square of the Euclidean distance between X and Pi . 6) are referred to as gravity problems (Francis and White 1974).