The Remainder of the Lecture Focuses on Developing a Ptas for 1jr J J ~

L max , a problem closely related to 1jr j jL max , which we will deene below. This will provide a rst example of a PTAS. In later lectures, we will exploit the technique of dynamic programming to develop other PTASs. 1 Approximation Schemes Finding a polynomial time optimization algorithm for a NP-hard optimization problem would imply that P = NP. Thus, given the widely held conjecture, that P 6 = NP, the best we can hope for is a method that provides arbitrarily good approximate solutions. Recall, the following two deenitions (made in the case of minimization problems): Deenition 1.1 A fully polynomial time approximation scheme (FPTAS) (for a given optimization problem) is an approximation algorithm with the following additional characteristics: 1. The algorithm's input is extended to include a nonnegative error threshold > 0 in addition to the input regarding the problem instance. 2. The algorithm produces a feasible solution whose value does not exceed (1 +)OPT, where OPT denotes the value of an optimal solution to the given problem instance. 3. The running time of the algorithm is polynomial in the length of the (original) input and in 1==. Deenition 1.2 A polynomial time approximation scheme (PTAS) (for a given optimization problem) is a collection fA g >0 of approximation algorithms, each of whose running times is polynomial in the length of the input, such that the algorithm A produces a feasible solution whose value does not exceed (1+)OPT, where OPT denotes the value of an optimal solution to the given problem instance.