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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
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 .


Source: Adler, Micah - Department of Computer Science, University of Massachusetts at Amherst


Collections: Computer Technologies and Information Sciences