Presentation is loading. Please wait.

Presentation is loading. Please wait.

Lecture 28 Approximation of Set Cover

Published byRoberta Conley Modified over 5 years ago

Similar presentations

Presentation on theme: "Lecture 28 Approximation of Set Cover"— Presentation transcript:

1 Lecture 28 Approximation 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

9 Hierarchy of Approximation

10 Theorem Proved using PCP system

11 MCDS Theorem (Guha-Khuller, 1998)

12

13 NP=P!

14

15 NP=P!

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

17 Theorem Connected Vertex-Cover has a 3-approximation.

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

19 Theorem

20

21

22 NP=P!

23

24

25

26 NP=P!

27

Download ppt "Lecture 28 Approximation of Set Cover"

Similar presentations

Algorithm Design Methods Spring 2007 CSE, POSTECH.

Algorithm Design Methods Spring 2007 CSE, POSTECH.
 Theorem 5.9: Let G be a simple graph with n vertices, where n>2. G has a Hamilton circuit if for any two vertices u and v of G that are not adjacent,

 Theorem 5.9: Let G be a simple graph with n vertices, where n>2. G has a Hamilton circuit if for any two vertices u and v of G that are not adjacent,
Optimization Problems in Optical Networks. Wavelength Division Multiplexing (WDM) Directed: Symmetric: Undirected: Optic Fiber.

Optimization Problems in Optical Networks. Wavelength Division Multiplexing (WDM) Directed: Symmetric: Undirected: Optic Fiber.
Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.

Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.
Set Cover 資工碩一 9562635 簡裕峰. Set Cover Problem 2.1 (Set Cover) Given a universe U of n elements, a collection of subsets of U, S ={S 1,…,S k }, and a cost.

Set Cover 資工碩一 簡裕峰. Set Cover Problem 2.1 (Set Cover) Given a universe U of n elements, a collection of subsets of U, S ={S 1,…,S k }, and a cost.
1 LP Duality Lecture 13: Feb 28. 2 Min-Max Theorems In bipartite graph, Maximum matching = Minimum Vertex Cover In every graph, Maximum Flow = Minimum.

1 LP Duality Lecture 13: Feb Min-Max Theorems In bipartite graph, Maximum matching = Minimum Vertex Cover In every graph, Maximum Flow = Minimum.
Approximation Algorithms

Approximation Algorithms
Design and Analysis of Algorithms Approximation algorithms for NP-complete problems Haidong Xue Summer 2012, at GSU.

Design and Analysis of Algorithms Approximation algorithms for NP-complete problems Haidong Xue Summer 2012, at GSU.
Bipartite Matching, Extremal Problems, Matrix Tree Theorem.

Bipartite Matching, Extremal Problems, Matrix Tree Theorem.
Lecture 9 Greedy Approach

Lecture 9 Greedy Approach
Approximating the Domatic Number Feige, Halldorsson, Kortsarz, Srinivasan ACM Symp. on Theory of Computing, pages 134-143, 2000.

Approximating the Domatic Number Feige, Halldorsson, Kortsarz, Srinivasan ACM Symp. on Theory of Computing, pages , 2000.
Lecture 21 Approximation Algorithms Introduction.

Lecture 21 Approximation Algorithms Introduction.
Optimization of Pearl’s Method of Conditioning and Greedy-Like Approximation Algorithm for the Vertex Feedback Set Problem Authors: Ann Becker and Dan.

Optimization of Pearl’s Method of Conditioning and Greedy-Like Approximation Algorithm for the Vertex Feedback Set Problem Authors: Ann Becker and Dan.
Approximating Maximum Edge Coloring in Multigraphs

Approximating Maximum Edge Coloring in Multigraphs
Perfect Graphs Lecture 23: Apr 17. Hard Optimization Problems Independent set Clique Colouring Clique cover Hard to approximate within a factor of coding.

Perfect Graphs Lecture 23: Apr 17. Hard Optimization Problems Independent set Clique Colouring Clique cover Hard to approximate within a factor of coding.
A general approximation technique for constrained forest problems Michael X. Goemans & David P. Williamson Presented by: Yonatan Elhanani & Yuval Cohen.

A general approximation technique for constrained forest problems Michael X. Goemans & David P. Williamson Presented by: Yonatan Elhanani & Yuval Cohen.
Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.

Is the following graph Hamiltonian- connected from vertex v? a). Yes b). No c). I have absolutely no idea v.
1 Bipartite Matching Lecture 3: Jan 17. 2 Bipartite Matching A graph is bipartite if its vertex set can be partitioned into two subsets A and B so that.

1 Bipartite Matching Lecture 3: Jan Bipartite Matching A graph is bipartite if its vertex set can be partitioned into two subsets A and B so that.
Is C 6 2-critical? a). Yes b). No c). I have absolutely no idea.

Is C 6 2-critical? a). Yes b). No c). I have absolutely no idea.
New Algorithm DOM for Graph Coloring by Domination Covering

New Algorithm DOM for Graph Coloring by Domination Covering

Similar presentations

Ads by Google