ApproximationAlgo.pdf - Is Close Enough Good Enough A-PDF Text Replace DEMO Purchase from www.A-PDF.com to remove the watermark Approximation Algorithms

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Is Close Enough Good Enough? Approximation Algorithms A-PDF Text Replace DEMO: Purchase from to remove the watermark 2 Design And Analysis of Algorithms Overview Definition Introduction Performance ratios Bin Packing The vertex-cover problem Traveling salesman problem Set cover problem 3 Design And Analysis of Algorithms Approximation algorithms Find an algorithm which return solutions that are guaranteed to be close to an optimal solution. Key: provably close to optimal Let OPT be the value of an optimal solution, and let SOL be the value of the solution that our algorithm returned. Constant factor approximation algorithms: SOL <= cOPT for some constant c. 4 Design And Analysis of Algorithms Introduction There are many important NP-Complete problems There is no fast solution ! But we want the answer … If the input is small use backtrack. Isolate the problem into P-problems ! Find the Near-Optimal solution in polynomial time. 5 Design And Analysis of Algorithms Performance ratios We are going to find a Near-Optimal solution for a given problem. We assume two hypothesis : Each potential solution has a positive cost. The problem may be either a maximization or a minimization problem on the cost. 6 Design And Analysis of Algorithms Performance ratios … ρ (n) If for any input of size n, the cost C of the solution produced by the algorithm is within a factor of ρ (n) of the cost C* of an optimal solution: Max ( C/C* , C*/C ) ρ (n) where C = Approximate cost and C* = Optimal cost We call this algorithm as an ρ (n)- approximation algorithm. 7 Design And Analysis of Algorithms  • • • 