Approximation Algorithms

Overview :
An approximation algorithm is a way of dealing with NP-completeness for an optimization problem. This technique does not guarantee the best solution. The goal of the approximation algorithm is to come as close as possible to the optimal solution in polynomial time. Such algorithms are called approximation algorithms or heuristic algorithms.

Features of Approximation Algorithm : 
Here, we will discuss the features of the Approximation Algorithm as follows.

• An approximation algorithm guarantees to run in polynomial time though it does not guarantee the most effective solution.
• An approximation algorithm guarantees to seek out high accuracy and top quality solution(say within 1% of optimum)
• Approximation algorithms are used to get an answer near the (optimal) solution of an optimization problem in polynomial time

Performance Ratios for approximation algorithms :
Here, we will discuss the performance ratios of the Approximation Algorithm as follows.

Scenario-1 :

1. Suppose that we are working on an optimization problem in which each potential solution has a cost, and we wish to find a near-optimal solution. Depending on the problem, we may define an optimal solution as one with maximum possible cost or one with minimum possible cost,i.e, the problem can either be a maximization or minimization problem.
2. We say that an algorithm for a problem has an appropriate ratio of P(n) if, for any input size n, the cost C of the solution produced by the algorithm is within a factor of P(n) of the cost C* of an optimal solution as follows.

max(C/C*,C*/C)<=P(n)

Scenario-2 :
If an algorithm reaches an approximation ratio of P(n), then we call it a P(n)-approximation algorithm.

• For a maximization problem, 0< C < C*, and the ratio of C*/C gives the factor by which the cost of an optimal solution is larger than the cost of the approximate algorithm.
• For a minimization problem, 0< C* < C, and the ratio of C/C* gives the factor by which the cost of an approximate solution is larger than the cost of an optimal solution.

Some examples of the Approximation algorithm :
Here, we will discuss some examples of the Approximation Algorithm as follows.

1. The Vertex Cover Problem –
   In the vertex cover problem, the optimization problem is to find the vertex cover with the fewest vertices, and the approximation problem is to find the vertex cover with few vertices.
   
2. Travelling Salesman Problem –
   In the traveling salesperson problem, the optimization problem is to find the shortest cycle, and the approximation problem is to find a short cycle.
   
3. The Set Covering Problem – 
   This is an optimization problem that models many problems that require resources to be allocated. Here, a logarithmic approximation ratio is used.
   
4. The Subset Sum Problem – 
   In the Subset sum problem, the optimization problem is to find a subset of {x1,×2,×3…xn} whose sum is as large as possible but not larger than the target value t.

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