Sampling in Graphs node sparsifiers Alexandr Andoni Microsoft

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

Sparsification of edges • Preserve some structure: e. g. , cuts • Also: distances,

Results on cut sparsifiers Graph size Approximation Reference Comments [Moi 09, LM 10, CLLM

Small cut (node) sparsifiers [A-Gupta-Krauthgamer’ 14] •

Main idea ? • Sampling edges doesn’t work here • Need to sample entire

Sampling in Bipartite Graphs • Sample non-terminals, together with edges • reweight edges accordingly

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Sampling in Graphs: node sparsifiers Alexandr Andoni (Microsoft Research)

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

Sparsification of edges • Preserve some structure: e. g. , cuts • Also: distances, effective resistances, etc

Sparsification of nodes ? •

Node sparsifiers •

Results on cut sparsifiers Graph size Approximation Reference Comments [Moi 09, LM 10, CLLM 10, EGKRTCT 10, MM 10] [LM 10, CLLM 10, MM 10] [Chu 12, KW 12] [HKNR 98, KRTV 12] [KRTV 12, KR 13] bipartite* graphs [AGK’ 14] bipartite* graphs • Similar results for flow (node) sparsifier

Small cut (node) sparsifiers [A-Gupta-Krauthgamer’ 14] •

Main idea ? • Sampling edges doesn’t work here • Need to sample entire sub-structures of the graph

Sampling in Bipartite Graphs • Sample non-terminals, together with edges • reweight edges accordingly

Sampling in Bipartite Graphs • Sample non-terminals, together with edges • reweight edges accordingly • Uniform sampling doesn’t work

Non-uniform sampling •