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This is because we can verify all decision versions of problems in P in polynomial time. 1 Example of Problems in NP For the class NP, we simply require that any “yes” answer is “easily” verifiable. In order to explain the verifier-based definition of NP, let us consider the subset sum problem: Assume there is a set of integers. The task of deciding whether a subset with sum zero exists is called the subset sum problem. Assume that we are given some integers, such as f 1; 2; 3; 9; 8g, and we wish to know whether some of these integers sum up to zero.

3) 40 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).

From one of the endpoints of this line, draw another contour line through the adjacent region with the corresponding slope. x; y/. x; y/ on the boundary of the region. There are four points about this algorithm. First, the numbers of vertical and horizontal lines need not be equal. Two facilities may have the same x coordinate but not the same y coordinate, thereby requiring one horizontal line and two vertical lines. In fact, this is why the index i of Vi ; ranges from one to p and that of Hi ranges from one to q.