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Unformatted text preview: Part VI: Dealing with Hard Problems Lecture 18: Approximation algorithms Lecture 18: Approximation algorithms Part VI: Dealing with Hard Problems Objective and Outline Objective: Introduction to approximate algorithms. Reference : Sections 35.1 & 35.2 of CLRS Outline: Introduction . Vertexcover problem. Travelingsalesman problem. Lecture 18: Approximation algorithms Part VI: Dealing with Hard Problems Introduction Many important problems are NPcomplete. What can we do if we have to somehow solve them? If input size is small, use exponential time algorithm . Identify special cases of the problems that can be solved in polynomial time. Try general optimization methods . e.g., branchandbound , genetic algorithms , neural nets . Quality of solution is not guaranteed in general. Sometimes, it is possible to find nearoptimal solutions. An approximation algorithm is one that returns an nearoptimal solution. Here we discuss some examples of approximation algorithms. Lecture 18: Approximation algorithms Part VI: Dealing with Hard Problems Performance ratios Suppose we work on an optimization problem where each solution has an associated a cost . An approximation algorithm returns a legal solution, but the cost of that legal solution may not be optimal. Examples: Suppose we are looking for a minimum size vertexcover (VC). An approximation algorithm returns a VC for us, but the size (cost) may not be minimum. We are looking for a maximum size independent set (IS). An approximation algorithm returns an IS for us, but the size (cost) may not be maximum. Lecture 18: Approximation algorithms Part VI: Dealing with Hard Problems Performance ratios Let C be the cost of the solution returned by an approximation algorithm, and C * be the cost of the optimal solution....
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 Spring '07
 ARYA
 Algorithms

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