1. of 17 Limits of Local Algorithms in Random Graphs Madhu Sudan MSR Joint work with David Gamarnik (MIT) 7/11/2013Local Algorithms on Random Graphs1

2. of 17 Main Result 7/11/2013Local Algorithms on Random Graphs2

3. of 17 Definition: Local Algorithms Informally: Local algorithms – Input = Communication network. – Wish to use local communication to compute some property of input. – In our case – large independent set in graph. – Allowed to use randomness, generated locally. 7/11/2013Local Algorithms on Random Graphs3

4. of 17 Formally 7/11/2013Local Algorithms on Random Graphs4

5. of 17 Locality in distributed algorithms – Usually algorithms try to compute some function of input graph, on the graph itself. – Algorithm uses data available topologically locally. – Leads to our model Locality a la Codes/Property Testing – Locality simply refers to number of queries to input. – More general model. – We can’t/don’t deal with it. 7/11/2013Local Algorithms on Random Graphs5

6. of 17 Motivations for our work 7/11/2013Local Algorithms on Random Graphs6

7. of 17 Motivations (contd.) 7/11/2013Local Algorithms on Random Graphs7

8. of 17 Proof 7/11/2013Local Algorithms on Random Graphs8

9. of 17 Clustering Phenomena 7/11/2013Local Algorithms on Random Graphs9

10. of 17 Clustering Theorem 7/11/2013Local Algorithms on Random Graphs10

11. of 17 7/11/2013Local Algorithms on Random Graphs11

12. of 17 Size of Ind. Set 7/11/2013Local Algorithms on Random Graphs12

13. of 17 7/11/2013Local Algorithms on Random Graphs13

14. of 17 7/11/2013Local Algorithms on Random Graphs14

15. of 17 Continuity (contd.) 7/11/2013Local Algorithms on Random Graphs15

16. of 17 Conclusions “Clustering” is an obstacle? Answer: – At least to local algorithms. – Local algorithms behave continuously, forcing non- clustering of solutions. Open questions: – Barrier to local algorithms in general sense? – To other complexity classes? 7/11/2013Local Algorithms on Random Graphs16

17. of 17 Thank You 7/11/2013Local Algorithms on Random Graphs17