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CSCI 3160 Design and Analysis of Algorithms

CSCI 3160 Design and Analysis of Algorithms. Yuanming Yu. Approximation algorithms. Motivation Some (optimization) problems are very hard Unlikely to have efficient polynomial-time algorithms Approximation algorithms Algorithms to find approximate solutions to optimization problems.

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CSCI 3160 Design and Analysis of Algorithms

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  1. CSCI 3160 Design and Analysis of Algorithms Yuanming Yu

  2. Approximation algorithms • Motivation • Some (optimization) problems are very hard • Unlikely to have efficient polynomial-time algorithms • Approximation algorithms • Algorithms to find approximate solutions to optimization problems.

  3. Optimization • Many problems are actually optimization problems • i.e – the task can be naturally rephrased as finding a maximal/minimal solution • For example: Vertex Cover problem

  4. Approximation Algorithm • An algorithm that returns near-optimalsolutions in polynomial time • Approximation Ratio ρ(n): • Define: C* as a optimal solution and C is the solution produced by the approximation algorithm • max (C/C*, C*/C) <= ρ(n) • Maximization problem: 0 < C <= C*, thus C*/C shows that C* is larger than C by ρ(n) • Minimization problem: 0 < C* <= C, thus C/C* shows that C is larger than C* by ρ(n)

  5. Techniques • A variety of techniques to design and analyze many approximation algorithms for computationally hard problems: • Combinatorial algorithms • Linear Programming based algorithms • Semi-definite Programming based algorithms

  6. Vertex Cover • Definition • Example

  7. Vertex Cover Problem • Definition: • Given an undirected graph G=(V,E), find a vertex cover with minimum size (has the least number of vertices) • This is sometimes called cardinality vertex cover • More generalization: • Given an undirected graph G=(V,E), and a cost function on vertices c: V → Q+ • Find a minimum cost vertex cover

  8. How to solve it • Matching: • A set M of edges in a graph G is called a matching of G if no two edges in set M have an endpoint in common • Example:

  9. How to solve it(cont.) • Maximum Matching: • A matching of G with the greatest number of edges • Maximal Matching: • A matching which is not contained in any larger matching • Note: Any maximum matching is certainly maximal, but not the reverse

  10. Main observation • No vertex can cover two edges of a matching • The size of every vertex cover is at least the size of every matching: |M| ≤ |C| • |M| = |C| indicates the optimality • Possible Solution: Using Maximal matching to find Minimum vertex cover

  11. An Algorithm • Alg: Find a maximal matching in G and output the set of matched vertices • Theorem: Algis a factor 2-approximation algorithm. • Proof:

  12. End • Questions?

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