Is Close Enough Good Enough?Approximation AlgorithmsA-PDF Text Replace DEMO: Purchase from to remove the watermark
2Design And Analysis of AlgorithmsOverview•Definition •Introduction•Performance ratios•Bin Packing•The vertex-cover problem•Traveling salesman problem•Set cover problem
3Design And Analysis of AlgorithmsApproximation 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.
4Design And Analysis of AlgorithmsIntroduction•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.
5Design And Analysis of AlgorithmsPerformance 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.
6Design And Analysis of AlgorithmsPerformance 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 costand C* = Optimal cost•We call this algorithm as an ρ(n)-approximation algorithm.