 
Summary: CMPSCI 611: Advanced Algorithms
Micah Adler
Problem Set 5 Out: November 30, 2000
Due: December 7 2000, 2000
1. [CLR] Problem 371 (page 983). For part (a), you can assume that the following Partition problem is
NPComplete:
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 NPComplete in the strong sense if the problem remains NPComplete even
when we restrict all numbers appearing in the input to be polynomial in the size of the input.
Suppose that a strongly NPComplete 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 .
