Homework Help is Here – Start Your Trial Now!
Engineering
Computer Science
There is an undirected graph G = (V = {v1, . . . , vn}, E). There are a total of n! possible orderings of these vertices. Pick one such ordering uniformly at random σ = (σ1, . . . , σn). Then consider the following process: Begin with S = ∅. Then, at each step (for i = 1 to n), if for all u ∈ S, (u, vσi ) ∈/ E, add vσi to S. Denote by d the maximum degree of a vertex of V . Prove that the proposed algorithm achieves an independent set with expected value of at least 1/d fraction of the optimal solution.

There is an undirected graph G = (V = {v1, . . . , vn}, E). There are a total of n! possible orderings of these vertices. Pick one such ordering uniformly at random σ = (σ1, . . . , σn). Then consider the following process: Begin with S = ∅. Then, at each step (for i = 1 to n), if for all u ∈ S, (u, vσi ) ∈/ E, add vσi to S. Denote by d the maximum degree of a vertex of V . Prove that the proposed algorithm achieves an independent set with expected value of at least 1/d fraction of the optimal solution.

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
See similar textbooks
icon
Related questions
Question

There is an undirected graph G = (V = {v1, . . . , vn}, E). There are a total of n! possible orderings
of these vertices. Pick one such ordering uniformly at random σ = (σ1, . . . , σn). Then consider the following process:
Begin with S = ∅. Then, at each step (for i = 1 to n), if for all u ∈ S, (u, vσi
) ∈/ E, add vσi
to S.
Denote by d the maximum degree of a vertex of V . Prove that the proposed algorithm achieves an independent set
with expected value of at least 1/d fraction of the optimal solution. 

Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Sets
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Database System Concepts
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS