Sampling in Graphs Alexandr Andoni (Microsoft Research)

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

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

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

Cut sparsifiers

Approach? [Karger’94,’96]:

Concentration

Applying Chernoff bound

Enough?

Smaller size?

Non-uniform sampling [Benczur-Karger’96]

Strong connectivity Connectivity: 5 Strong conn.: 2

Proof of theorem

ii) Cut values are approximated

Iterative sampling

Comments

BREAK

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

Motivating example

Proof of theorem

i) Sketch description

ii) Sketch size ???

iii) Estimation

Estimation illustration dense components

iii) Correctness of estimation

Variance

Dense component estimate

Concluding remarks

Open questions