Approximation Algorithm
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Transcript of Approximation Algorithm
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Approximation Algorithm
Prepared by:Lamiya El_Saedi
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Introduction:
There are many hard combinatorial optimization problems that can’t be solved efficiently using backtracking or randomization.
The alternative way for talking some of these problem is to devise an approximation algorithm.
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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.
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A marking characteristic of approximation algorithms is that they are fast, as they are mostly greedy heuristics.
The proof of correctness of greedy algorithm may be complex.
In general, the better the performance bound the harder it becomes to prove the correctness of an approximation algorithms.
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Basic Definition:
Combinatorial optimization
problem
A set DII of instances
For each I in DII
There is SII(I) of Candidate solution
For each solution σ
In SII(I) there is
A value fII(σ)Called the solution
value of σ
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Note:
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Cont.
In simple word: assume that: DII={I1,…,In} SII(Ii)={σ1,…, σn} fII(σi)={v1,…,vn} fII(σ)=A(I)
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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.
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Subset-sum algorithm:
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Cont.
Time complexity of algorithm is exactly the size of the table Θ(nC) as filling each entry requires Θ(1) time.
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Cont. When I apply the example by using subset-
sum algorithm the results appear like this:
012345
0000000
s1011111
s2012333
s3012345
s4012345
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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
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Now:
We develop an approximation algorithm
for some positive integer k.
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