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Lecture 27 Set Cover & LP-Relaxation

Lecture 27 Set Cover & LP-Relaxation. Ding-Zhu Du. Set-Cover. Given a collection C of subsets of a set E , find a minimum subcollection C’ of C such that every element of E appears in a subset in C’. Potential Function. Greedy Algorithm. Analysis. Proof of Inequality.

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Lecture 27 Set Cover & LP-Relaxation

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  1. Lecture 27Set Cover & LP-Relaxation Ding-Zhu Du

  2. Set-Cover Given a collection C of subsets of a set E, find a minimum subcollection C’ of C such that every element of E appears in a subset in C’ .

  3. Potential Function

  4. Greedy Algorithm

  5. Analysis

  6. Proof of Inequality

  7. What’s we need?

  8. Actually, this inequality holds if and only if f is submodular and (monotone increasing)

  9. Rounding • Combinatorial Rounding vertex-cover scheduling on unrelated parallel machine • Random Rounding Maximum satisfiability

  10. Vertex Cover

  11. LP-Relaxation Rounding Theorem Proof.

  12. Half-integerality Theorem Proof

  13. Max Sat

  14. LP-relaxation

  15. Random Rounding

  16. Calculation of E(Z)

  17. Lower bound of E(Zj) Lemma Proof

  18. 0 1

  19. Performance

  20. Derandomization Theorem

  21. Thanks, End

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