Fully Poly-Time Approximation Scheme

• Returned value is largest < t to within some percentage error
• Let $\epsilon$ = error bound, 0 < $\epsilon$ < 1
• Trim L of all values whose relative error is no more then $\delta$ away from a value in L

Trim(L, $\delta$) $\;\;\;\;\;$ $\;\;\;\;\;$; L = <y₁, .., y_m> sorted in non-decreasing order
1 $\;\;\;\;\;$m =
2 $\;\;\;\;\;$L' =
3 $\;\;\;\;\;$last = y₁
4 $\;\;\;\;\;$for i = 2 to m
5 $\;\;\;\;\;$ $\;\;\;\;\;$if last < (1 - $\delta$)y_i
6 $\;\;\;\;\;$ $\;\;\;\;\;$then append y_i on end of L'
7 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$last = y_i
8 $\;\;\;\;\;$return L'

Approx-Subset-Sum(S, t, $\epsilon$)
1 $\;\;\;\;\;$n = $\mid$S$\mid$
2 $\;\;\;\;\;$L₀ = $\langle$0$\rangle$
3 $\;\;\;\;\;$for i = 1 to n
4 $\;\;\;\;\;$ $\;\;\;\;\;$L_i = Merge-Lists( )
5 $\;\;\;\;\;$ $\;\;\;\;\;$L_i = Trim(L_i, )
6 $\;\;\;\;\;$ $\;\;\;\;\;$remove from L_i elements > t
7 $\;\;\;\;\;$return largest value in L_n

Error passed to Trim is to prevent too much inaccuracy after repeated trimmings.

Theorem 37.6

Approx-Subset-Sum is a fully poly-time approximation scheme for the subset-sum problem.