1 Approximation Through Scaling Algorithms and Networks 2014/2015 Hans L. Bodlaender Johan M. M. van Rooij

2 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)

3 PTAS  A Polynomial Time Approximation Scheme is an algorithm, that gets two inputs: the “usual” input X, and a real value >0.  For each fixed >0, the algorithm  Uses polynomial time  Is an (1+)-approximation algorithm

4 FPTAS  A Fully Polynomial Time Approximation Scheme is an algorithm, that gets two inputs: the “usual” input X, and a real value >0, and  For each fixed >0, the algorithm  Is an (1+)-approximation algorithm  The algorithm uses time that is polynomial in the size of X and 1/.

5 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.

6 Dynamic Programming for Knapsack  Let P be the maximum value of all items.  We can solve the problem in O(n 2 P) 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) }

7 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) = v(i) / c   Solve scaled instance with DP optimally  Output this solution: approximates solution for original instance

8 The Question is….  How do we set c, such that  Approximation ratio good enough  Polynomial time

9 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

10 Setting c  Set c = P/(2n)  This is a FPTAS  Running time:  Largest value of an item is at most P/c = P / (P/(2n)) = 2n/.  Running time is O(n 2 * 2n/) = O(n 3 /)  Approximation: … next

11 -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( P / (2n) ) = OPT – /2 P  OPT / (OPT – /2 P)≤ OPT / (OPT – /2 OPT) = 1/(1-/2) ≤ 1+

12 Comments  Technique works for many problems .. But not always …