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

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PTAS(Polynomial Time Approximation Scheme) cont.

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## PTAS(Polynomial Time Approximation Scheme) cont.

Prepared by, Umair S.

March 25th, 2009

### 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

• Input

• Output

### 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,

### 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)

### 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…