Approximation Algorithms for Knapsack Problems 1 Tsvi Kopelowitz Modified by Ariel Rosenfeld

Knapsack Given: a set S of n objects with weights and values, and a weight bound: ◦w 1, w 2, …, w n, B (weights, weight bound). ◦v 1, v 2, …, v n (values - profit). Find: subset of S with total weight at most B, and maximum total value. Formally: Problem is known to be NP-hard 2

Assumptions. (every item can be added to T). (non-negative values) Values, weights, and bound are all integers. Note: This is a maximum problem. Define: OPT = The optimal solution. We will see a 2-approximation for two versions of knapsack. 3

Uniform Knapsack (simple form) (value=weight) 2-approximation algorithm: ◦Sort the items such that v 1 ≥ v 2 ≥ … ≥ v n. ◦Pick such that and. 4

Uniform Knapsack (proff) Claim:. Proof: ◦Assume (by contradiction):. ◦By the algorithm: =>. ◦ => =>contradiction. (Since items were sorted) 5

2-approximation (general knapsack) Define ratio: Sort items such that r 1 ≥ r 2 ≥ … ≥ r n. Pick such that and. If( ) return else return A. 6

2-approximation try it yourself… 7

Claim: Proof sketch (fill the details yourselves): Consider fractional knapsack (where we can take some of each item)– the optimal solution is A’= (A and a fraction of v j+1.) ◦. ◦.◦. 2-approximation (general knapsack) 8

9 A(i,j) = Smallest weight subset of objects 1,…,i with a total value of j. A DP algorithm for knapsack A j n v max i n Upper bound on optimal profit

A DP algorithm for knapsack The result is: max j such that A(n, j) ≤ B. The runtime is: O(n 2 v max ) Pseudo-polynomial 10

11 Definition: Fully Polynomial Time Approximation Scheme (FPTAS) Given ε, delivers a solution with a ratio of (1- ε ) for maximum and a ratio of (1+ ε ) for minimum, and runs in time polynomial in the size of the input and (1/ ε ) Definition: Pseudo-polynomial If input of integers is given in unary form, runs in polynomial time. Definitions

FPTAS for knapsack The Idea – use scaling!! Given error bound, define. For each object i, define. Use DP to find optimal solution OPT’ for the rounded values. Return the set OPT’ with the original values. 12

Correctness Claim: Proof: For every i:. (1) => (3) (2) (4) 13

Correctness Proof continued: (1) (3) (2) (4) 14

Complexity and Notes Time: algorithm is a FPTAS. 15