Approximation Through Scaling
Algorithms and Networks 2014/2015
Hans L. Bodlaender
Johan M. M. van Rooij
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C-approximation
 Optimization problem: output has a value that we want to
maximize or minimize
 An algorithm A is an C-approximation algorithm, if for all
inputs X, we have that (with OPT the optimal value)
 value(A(X)) / OPT ≤ C (minimisation), or
 OPT / value(A(X)) ≤ C (maximisation)
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PTAS
 A Polynomial Time Approximation Scheme is an algorithm,
that gets two inputs: the “usual” input X, and a real value
e>0.
 For each fixed e>0, the algorithm
 Uses polynomial time
 Is an (1+e)-approximation algorithm
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FPTAS
 A Fully Polynomial Time Approximation Scheme is an
algorithm, that gets two inputs: the “usual” input X, and a
real value e>0, and
 For each fixed e>0, the algorithm
 Is an (1+e)-approximation algorithm
 The algorithm uses time that is polynomial in the size of X
and 1/e.
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Example: Knapsack
 Given: n items each with a positive integer weight w(i) and
integer value v(i) (1 ≤ i ≤ n), and an integer B.
 Question: select items with total weight at most B such
that the total value is as large as possible.
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Dynamic Programming for Knapsack
 Let P be the maximum value of all items.
 We can solve the problem in O(n2P) time with dynamic
programming:
 Tabulate M(i,Q): minimum total weight of a subset of items 1, …, i
with total value Q for Q at most nP
 M(0,0) = 0
 M(0,Q) = ∞, if Q>0
 M(i+1,Q) = min{ M(i,Q), M(i,Q-v(i+1)) + w(i+1) }
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Scaling for Knapsack
 Take input for knapsack
 Throw away all items that have weight larger than B (they
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
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are never used)
Let c be some constant
Build new input: do not change weights, but set new
values v’(i) = v(i) / c 
Solve scaled instance with DP optimally
Output this solution: approximates solution for original
instance
The Question is….
 How do we set c, such that
 Approximation ratio good enough
 Polynomial time
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Approximation ratio
 Consider optimal solution Y for original problem, value OPT
 Value in scaled instance: at least OPT/c – n
 At most n items, for each v(i)/c - v(i)/c  <1
 So, DP finds a solution of value at least OPT/c –n for scaled
problem
 So, value of approximate solution for original instance is at
least c*(OPT/c –n) = OPT - nc
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Setting c
 Set c = eP/(2n)
 This is a FPTAS
 Running time:
 Largest value of an item is at most P/c = P / (eP/(2n)) = 2n/e .
 Running time is O(n2 * 2n/e) = O(n3/e)
 Approximation: … next
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e-approximation
 Note that each item is a solution (we removed items with
weight more than B).
 So OPT ≥ P.
 Algorithm gives solution of value at least:
OPT – nc = OPT – n(e P / (2n) ) = OPT – e/2 P
 OPT / (OPT – e/2 P) ≤ OPT / (OPT – e/2 OPT)
= 1/(1-e/2) ≤ 1+e
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Comments
 Technique works for many problems
 .. But not always …
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