1. Sampling in Graphs Alexandr Andoni (Microsoft Research)

2. Graph compression Why smaller graphs? use less storage space faster algorithms easier visualization

3. Preserve some structure Cuts approximately Other properties: Distances, (multi-commodity) flows, effective resistances…

4. Plan 1) Cut sparsifiers 2) More efficient cut sparsifiers 3) Node sparsifiers

5. Cut sparsifiers

6. Approach? [Karger’94,’96]:

9. Concentration

10. Applying Chernoff bound

11. Enough?

12. Smaller size?

13. Non-uniform sampling [Benczur-Karger’96]

14. Strong connectivity Connectivity: 5 Strong conn.: 2

15. Proof of theorem

16. ii) Cut values are approximated

17. Iterative sampling

18. Comments

19. BREAK

21. Smaller relaxed cut sparsifiers [A-Krauthgamer-Woodruff’14]:

22. Motivating example

23. Proof of theorem

24. i) Sketch description

25. ii) Sketch size ???

26. iii) Estimation

27. Estimation illustration dense components

28. iii) Correctness of estimation

30. Variance

31. Dense component estimate

32. Concluding remarks

33. Open questions