18 - Approximate Algorithms

18 - Approximate Algorithms - Part VI: Dealing with Hard...

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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 . Vertex-cover problem. Traveling-salesman problem. Lecture 18: Approximation algorithms Part VI: Dealing with Hard Problems Introduction Many important problems are NP-complete. 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., branch-and-bound , genetic algorithms , neural nets . Quality of solution is not guaranteed in general. Sometimes, it is possible to find near-optimal solutions. An approximation algorithm is one that returns an near-optimal 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 vertex-cover (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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18 - Approximate Algorithms - Part VI: Dealing with Hard...

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