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Fully PTAS Knapsack Decision Version Proof Theorem Algorithm

Learn about the proof and theorem of the Fully PTAS for the Knapsack Decision Version algorithm, its complexity, and the classification and implementation details.

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Fully PTAS Knapsack Decision Version Proof Theorem Algorithm

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  1. Lecture 25 Partition

  2. Partition

  3. Subsum

  4. S=33333333333333333333 1111111111111111111111 1 1

  5. Puzzle Answer

  6. Knapsack

  7. Decision Version

  8. Theorem Proof.

  9. Theorem

  10. Algorithm • Classify: for i < m, ci< a= cG, for i > m+1, ci > a. • Sort • For

  11. Proof.

  12. Time

  13. Fully PTAS • A problem has a fully PTAS if for any ε>0, it has (1+ε)-approximation running in time poly(n,1/ε).

  14. Fully FTAS for Knapsack

  15. Time • outside loop: O(n) • Inside loop: O(nM) where M=max ci • Core: O(n log (MS)) • Total O(n M log (MS)) • Since input size is O(n log (MS)), this is a pseudo-polynomial-time due to M=2 3 log M

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