# L17-PTAS - Polynom tim ial e approxim ation sche e m...

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Polynomial time Polynomial time approximation scheme approximation scheme Lecture 17: Mar 13

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Polynomial Time Approximation Scheme (PTAS) Polynomial Time Approximation Scheme (PTAS) We have seen the definition of a constant factor approximation algorithm. The following is something even better. An algorithm A A is an approximation scheme if for every є > 0, A runs in polynomial time (which may depend on є ) and return a solution: SOL (1+ є )OPT for a minimization problem SOL (1- ≥ є )OPT for a maximization problem For example, A A may run in time n 100/ є . There is a time-accuracy tradeoff.
Knapsack Problem Knapsack Problem A set of items, each has different size and different value. We only have one knapsack. Goal: to pick a subset which can fit into the knapsack and maximize the value of this subset.

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Knapsack Problem Knapsack Problem (The Knapsack Problem) Given a set S = {a 1 , …, a n } of objects, with specified sizes and profits, size(a i ) and profit(a i ), and a knapsack capacity B, find a subset of objects whose total size is bounded by B and total profit is maximized. Assume size(a i ), profit(a i ), and B are all integers. We’ll design an approximation scheme for the knapsack problem.
Greedy Methods Greedy Methods Sort by object size in non-decreasing order: General greedy method : Sort the objects by some rule, and then put the objects into the knapsack according to this order. Sort by profit in non-increasing order: Sort by profit/object size in non-increasing order: Greedy won’t work.

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Exhaustive Search Exhaustive Search n objects, total 2 n possibilities, view it as a search tree. choose object 1 not choose object 1 choose object 2 not choose object 2 choose object 2 not choose object 2 At the bottom we could calculate the total size and total profit, and choose the optimal subset.
Exhaustive Search Exhaustive Search size(a1)=2, profit(a1)=4 size(a2)=3, profit(a2)=5 size(a3)=2, profit(a3)=3 size(a4)=1, profit(a4)=2 (0,0) (Total size, total profit) choose object 1 not choose object 1 (2,4) (0,0) (0,0) (0,0) (5,9) (2,4) (2,4) (4,7) (5,9) (7,12) (3,5) (3,5) (5,8) (2,3) There are many redundancies.

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Exhaustive Search Exhaustive Search size(a1)=2, profit(a1)=4 size(a2)=3, profit(a2)=5 size(a3)=2, profit(a3)=3 size(a4)=1, profit(a4)=2 (0,0) (Total size, total profit) choose object 1 not choose object 1 (2,4) (0,0) (0,0) (0,0) (5,9) (2,4) (2,4) (4,7) (5,9) (7,12) (3,5) (3,5) (5,8) (2,3) There are many redundancies. (4,7) (5,9) (5,9) (6,11) (4,7) (3,5)
Exhaustive Search size(a1)=2, profit(a1)=4 size(a2)=3, profit(a2)=5 size(a3)=2, profit(a3)=3 size(a4)=1, profit(a4)=2 (0,0) (Total size, total profit) choose object 1 not choose object 1 (2,4) (0,0) (0,0) (0,0) (5,9) (2,4) (2,4) (4,7) (5,9) (7,12) (3,5) (3,5) (5,8) (2,3) There are many redundancies. (4,7)

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L17-PTAS - Polynom tim ial e approxim ation sche e m...

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