Approximation Algorithm

Approximation Algorithm Prepared by: Lamiya El_Saedi

Introduction: • There are many hardcombinatorial optimization problemsthat can’t be solved efficiently using backtracking or randomization. • The alternative way for talking some of these problem is to devise an approximation algorithm.

The approximation is depend on the reasonable solution that approximations as optimal solution • There is a performance bound that guarantees that the solution to a given instance will not be far away from the neighborhood of the exact solution.

A marking characteristic of approximation algorithms is that they arefast, as they are mostlygreedy heuristics. • The proof of correctness ofgreedy algorithm may be complex. • In general, the better the performance bound the harder it becomes to prove the correctness of an approximation algorithms.

Basic Definition:

Note:

Cont. • In simple word: assume that: DII={I1,…,In} SII(Ii)={σ1,…, σn} fII(σi)={v1,…,vn} fII(σ)=A(I)

Subset-sum problem: • Is a special case of the Knapsack problem in which the item values are identical to their sizes. • Ex: I= {I1,I2,I3,I4} S= {1,2,3,4} V= {1,2,3,4} C (Knapsack capacity)= 5 • The objective is to find a subset of the items that maximizes the total sum of their sizes without exceeding the Knapsack capacity.

Subset-sum algorithm:

Cont. • Time complexity of algorithm is exactly the size of the table Θ(nC) as filling each entry requires Θ(1) time.

Cont. • When I apply the example by using subset-sum algorithm the results appear like this:

Cont. • So, from the table: OPT(4)={1} <4 OPT(3)={1,2} <3 OPT(2)={0} <2 does not exist in DII OPT(1)= {0} <1 does not exist in DII

Now: • We develop an approximation algorithm for some positive integer k.

Approximation Algorithm