1. PTAS(Polynomial Time Approximation Scheme) cont. Prepared by, Umair S. March 25 th, 2009

2. PTAS vs FPTAS  PTAS requires the complexity of an algorithm to be polynomial in terms of input size n for a fixed approximation factor є  FPTAS requires the complexity of an algorithm to be polynomial, both in terms of n as well as 1/ є

3. Designing Polynomial Time Approximation Scheme for Sub-set Sum Problem  Input  Output

4. Designing Polynomial Time Approximation Scheme for Sub-set Sum Problem  In case of approximation, we are interested in a S’ such that  We define, L i be the set of numbers that are sum of all elements in each possible subsets of set S i where, S i is a set of first i th elements in set S. Then,

5. Designing Polynomial Time Approximation Scheme for Sub-set Sum Problem  Pseudo-code for finding the closest sub-sum can be  While i<n  Remove where, l j is any element in set L i  end while  Solution: last element of Ln  Complexity: O(nW)

6. Designing Polynomial Time Approximation Scheme for Sub-set Sum Problem  Complexity is O(nW), W can be exponential in the worst-case!  Consider small intervals instead of exact values in Li?  Equally spaced vs expanding intervals?  Possible to maintain an approximation factor? To be cont. in next lecture…