PTAS(Polynomial Time Approximation Scheme) cont.

1. PTAS(Polynomial Time Approximation Scheme) cont. Prepared by, Umair S. March 25th, 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, Libe the set of numbers that are sum of all elements in each possible subsets of set Si where, Si is a set of first ith 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, lj is any element in set Li • 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…