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Approximation Through Scaling

Algorithms and Networks 2014/2015Hans 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

are never used) Let c be some constant Build new input: do not change weights, but set new values

v’(i) = ë v(i) / c û Solve scaled instance with DP optimally Output this solution: approximates solution for original

instance

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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 - ë v(i)/c û <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 …