Towards 1Approximate Flow Sparsifiers Robert Krauthgamer Weizmann Institute

Towards (1+ϵ)-Approximate Flow Sparsifiers Robert Krauthgamer, Weizmann Institute of Science Joint work with Anupam Gupta and Alexandr Andoni Dagstuhl, April 2017

Graph Sparsification n Vast literature on “compression” (succinct representation) of graphs q We focus on preserving specific features – distances, cuts, etc. exactly/approximately n Edge sparsification: q q Cut and spectral sparsifiers [Benczur-Karger, Spielman-Teng, …] Spanners [Peleg-Ullman, …] or mostly n Vertex sparsification (keep only the “terminal” vertices) q q Cut/multicommodity-flow sparsifier [Hagerup-Katajainen-Nishimura. Ragde, Moitra, …] Distances [Gupta, Coppersmith-Elkin, …] Towards (1+ϵ)-Approximate Flow Sparsifiers

Terminal Cuts n n Network G with edge capacities c: E(G) R+. k terminals K½V(G) (“huge” network) (“important” vertices) G: n We care about terminal cuts: q mincut. G(S) = minimum-capacity cut separating S½K and S=Kn. S. q (Equivalent to the maximum flow between S and S. ) Towards (1+ϵ)-Approximate Flow Sparsifiers

Mimicking Networks n A mimicking network of (G, c) is a network (G’, c’) with same terminals and 8 SµK, mincut. G(S) = mincut. G’(S). G: 3 9 A B exact 2 5 1 8 C G’: A 3 B 1 C 4 n Theorem [Hagerup-Katajainen-Nishimura-Ragde’ 95]. Every k-terminal network has a mimicking network of · 22 k vertices. q q q n Pro: independent of n=|V(G)| Con: more wasteful than listing the 2 k cut values (Originally proved for directed networks) Intuition: There are · 2 k relevant cuts (choices for S), which jointly partition the vertices to · 22 k “buckets”; merge each bucket … Towards (1+ϵ)-Approximate Flow Sparsifiers

Approximate Vertex-Sparsifiers n Towards (1+ϵ)-Approximate Flow 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, KR 14] [KRTV 12, KR 13] here bipartite* graphs here Towards (1+ϵ)-Approximate Flow Sparsifiers series-parallel

Results on Cut Sparsifiers Graph size flow 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, KR 14] [KRTV 12, KR 13] here bipartite* graphs here series-parallel here is a sketch Towards (1+ϵ)-Approximate Flow Sparsifiers

Our Results n Theorem. Bipartite* networks admit flow sparsifiers of quality 1+² and size poly(k/²) q Bipartite* = the non-terminals form an independent set Bypasses 2 (k) bound we saw for exact sparsifiers (even in bipartite) q This talk: cut sparsifiers (simpler, but extends to flow sparsifiers) q Towards (1+ϵ)-Approximate Flow Sparsifiers

Main Idea: Structure Sampling n n Edge sampling useful for edge-sparsifiers [BK’ 96, SS’ 11] But does not work here, need to sample entire sub-structures Towards (1+ϵ)-Approximate Flow Sparsifiers

Sampling in Bipartite Graphs n Sample non-terminal vertices, together with incident edges q reweight edges accordingly Towards (1+ϵ)-Approximate Flow Sparsifiers

Sampling in Bipartite Graphs n Sample non-terminal vertices, together with incident edges q n reweight edges accordingly Uniform sampling does not work Towards (1+ϵ)-Approximate Flow Sparsifiers

Non-Uniform Sampling n Towards (1+ϵ)-Approximate Flow Sparsifiers

n Towards (1+ϵ)-Approximate Flow Sparsifiers

Importance Sampling n Towards (1+ϵ)-Approximate Flow Sparsifiers

Actual Sampling n Towards (1+ϵ)-Approximate Flow Sparsifiers

Open Questions n Extend to general networks? q q q n Want to beat size 22 k (exact sparsification) Need to sample other structures (flow paths? ? ) Handle planar networks? What about flow-sparsifiers? q q q In bipartite networks: (our technique extends) In general networks: no bound f(k, ²) is known A positive indication: can build a data structure of size ≈(1/²)k 2 (“big table” with all values) Thank You! Towards (1+ϵ)-Approximate Flow Sparsifiers