CMPSCI 611: Advanced Algorithms Micah Adler Summary: CMPSCI 611: Advanced Algorithms Micah Adler Problem Set 5 Out: November 30, 2000 Due: December 7 2000, 2000 1. [CLR] Problem 37-1 (page 983). For part (a), you can assume that the following Partition problem is NP-Complete: INPUT: a set of positive integers A = fa 1 : : : an g. QUESTION: can the set A be partitioned into two sets S and A S such that the sum of the integers in S is equal to the sum of the integers in A S? 2. We say that there is a fully polynomial time approximation scheme (FPTAS) for a problem if there is an approximation algorithm that takes as input an instance of the problem and a value  > 0, and returns a solution that is within a factor of 1 +  of optimal. The running time of the algorithm must be polynomial in the size of the input, as well as in 1  . We say that a problem is NP-Complete in the strong sense if the problem remains NP-Complete even when we restrict all numbers appearing in the input to be polynomial in the size of the input. Suppose that a strongly NP-Complete maximization problem has the property that for all inputs x, the optimum cost is bounded by p(NUM(x)), where p() is a polynomial, and NUM(x) is the largest number appearing in the input x. Show that if there is a FPTAS for such a problem, then P = NP . Collections: Computer Technologies and Information Sciences