1. The Subset-sum Problem
Problem statement
given the set S = {x1, x2, x3, … xn } of positive integers and t, is there a subset of S that adds up to t
as an optimization problem, what subset of S adds up to the greatest total <= t
What we will show
first we develop an exponential time algorithm to solve the problem exactly
we modify this algorithm to give us a fully polynomial time approximation scheme that has a running time that is polynomial in 1/e and n

2. The exponential algorithm
Some preliminaries
we use the notation S + x = { s+x : s  S}
we have a merge lists algorithm that runs in time proportional to the sum of the lengths of the two lists

3. Why it works Some notation Proof and complexity
Pi is the set of sums of all subsets of values up to xi
it can be shown that
Proof and complexity
by induction on i it can be shown that Li is a sorted list that contains every element of Pi <= t
since Li can be as long as 2i, the algorithm, in general, is exponential
polynomial time algorithms exist if t is polynomial in |S| or all numbers in S are bounded by a polynomial in |S|

4. Trimming a List
To trim by a factor d means a term z can be represented by a smaller term provided it is “close enough”
The complexity is (m)

5. A fully-polynomial scheme
The algorithm is similar to our exponential algorithm except we now perform a trim step

6. An example