300 likes | 313 Views
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.
E N D
Lecture 21MoreApproximation Algorithms Introduction
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
Max Coverage Red + Green
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, for i > m+1, ci > a. • Sort • For
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).
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)
CS6382 CS7301-CS6301