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By Bernard Moret, Henry D. Shapiro

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2 Satisfiability Problem Given: A Boolean expression – a function of true/false variables. Question: Is there an assignment of truth values (TRUE or FALSE) to the variables such that the expression is TRUE (Daskin 1995). As the problem shows, satisfiability (SAT) problem is essentially expressed in the form of a decision problem. We just need to acquire the answer “yes” or “no”. A Boolean function is a function whose variable values and function value are all in fTRUE, FALSEg. We often denote TRUE by 1 and FALSE by 0 (Du and Ko 2000).

The number of such paths is, of course, infinite (Francis and White 1974). x; y/ D Min m X wi jx i D1 ai j C m X wi jy i D1 Pi X Fig. 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.

6 Reduction A reduction is a way of converting one problem into another problem in such a way that, if the second problem is solved, it can be used to solve the first problem (Sipser 1996). For example, suppose you want to find your way around a new city. You know this would be easy if you had a map. This demonstrates reducibility. The problem of finding your way around the city is reducible to the problem of obtaining a map of the city (Sipser 1996). Many examples also can be found in mathematics.

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