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Lecture 28 Approximation of Set Cover

Lecture 28 Approximation of Set Cover. Min Set Cover. Red + Green. Greedy Algorithm. Observation. Theorem. Theorem. Greedy Algorithm produces an approximation within ln n +1 from optimal. The same result holds for weighted set-cover. Theorem. Proof. Hierarchy of Approximation.

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Lecture 28 Approximation of Set Cover

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  1. Lecture 28Approximation of Set Cover

  2. Min Set Cover Red + Green

  3. Greedy Algorithm

  4. Observation

  5. Theorem

  6. Theorem Greedy Algorithm produces an approximation within ln n +1 from optimal. The same result holds for weighted set-cover.

  7. Theorem Proof.

  8. Hierarchy of Approximation

  9. Theorem Proved using PCP system

  10. MCDS Theorem (Guha-Khuller, 1998)

  11. NP=P!

  12. NP=P!

  13. Connected Vertex-Cover • Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.

  14. Theorem • Connected Vertex-Cover has a 3-approximation.

  15. Weighted Connected Vertex-Cover Given a vertex-weighted connected graph, find a connected vertex-cover with minimum total weight. TheoremWeighted Connected Vertex-Cover has a (1+ln Δ)-approximation where Δ is the maximum node degree of input graph. This is the best-possible!!!

  16. Theorem

  17. NP=P!

  18. NP=P!

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