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Efficient Approximation Algorithms for Combinatorial Optimization Problems

Many hard combinatorial optimization problems cannot be efficiently solved through backtracking or randomization techniques. Instead, approximation algorithms provide a viable alternative, offering reasonable solutions that approach optimality. These algorithms are characterized by their fast and mostly greedy heuristics, although proving their correctness can be complex. This text explores the principles of approximation algorithms, their performance bounds, and includes an example of the subset-sum problem, emphasizing their practical applications in optimization.

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Efficient Approximation Algorithms for Combinatorial Optimization Problems

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  1. Approximation Algorithm Prepared by: Lamiya El_Saedi

  2. 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.

  3. 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.

  4. 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.

  5. Basic Definition:

  6. Note:

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

  8. 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.

  9. Subset-sum algorithm:

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

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

  12. 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

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

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