Lecture 21 More Approximation Algorithms Introduction

Maximum 3DM

3-Approximation Any maximal 3DM is a 3-approximation for max 3DM. This is because in the maximum 3DM, every edge (3-set) must have at least one vertex covered by the maximal 3DM.

Min Set Cover Red + Green

Greedy Algorithm

Observation

Theorem

Max Coverage Red + Green

Greedy Algorithm

Theorem

Lower Bound

Knapsack

2-approximation

PTAS A problem has a PTAS (polynomial-time approximation scheme) if for any ε > 0, it has a (1+ε)-approximation.

Knapsack has PTAS Classify: for i < m, ci < a= cG, Sort For

Proof.

Time

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).

Fully FTAS for Knapsack

Pseudo Polynomial-time Algorithm for Knapsak Initially,

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

Complexity of Approximation FPTAS (e.g., Knapsack) PTAS (e.g., Knapsack) Constant-approximation (e.g., vertex-cover) -approximation (e.g., set cover) -approximation (e.g., max clique)

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