What have we learnt about graph expansion in the new millenium ?

What have we learnt about graph expansion in the new millenium? Sanjeev Arora Princeton University &Center for Computational Intractability

Overview • Last millenium: • Central role of expansion and expanders • Recognizing expander graphs viaeigenvalues(Cheeger71,Alon-Milman85) • O(log n)-approximation via flows(Leighton-Rao88);region-growing technique; • Close connection to metric embeddings; O(log n) approximation for generalsparsest cut via Bourgain’s Embedding Theorem (Linial-London-Rabinovich, Aumann-Rabani) • This millenium (so far): • O(√log n )-approximation (A., Rao, Vazirani 04) via both SDPand flows; • Better metric embeddings; O(√log n )-approximation for general sparsest cut (Chawla-Gupta-Raecke05, A.-Lee-Naor06) • Inapproximability results via Unique Games Conjecture (CKKRS06; KV06) • Lowerboundsfor metric embeddings (inspired by PCPs) [KV06; others]; lowerbounds for SDPs; • Progress in relating full eigenvalue spectrum to (small-set) expansion (A., Barak, Steurer10) (Will not talk about: New understanding of expansion in Cayley graphs of groups, newalgebra-free constructions of optimal expanders, etc.)

Graph Expansion Important concept:derandomization, network routing, coding theory, Markov chains, differential geometry, group theory d-regular graph G d expansion(S) = vertex set S # edges leaving S d |S| a-expander: expansion(S) ≥ a for all S (co-NP-hard to recognize) α2/2 ≤ λ ≤ 2α [Cheeger, Alon85, Alon-Milman85]. λ= smallest nonzero eigenvalueof Laplacianof G.Allows us to recognizegraphs with α = Ω(1)(“expander”) (often will restrict attention wlog to “balanced” sets: |S|, |Sc| > W(n))

Approximating expansion via flows [Leighton-Rao’88] O(log n)-approximation.Find largestbs.t. we can simultaneously route b/nunits offlow between every vertex pair.(“embed a complete graph”)

Why α ≥ β/2 : Total flow out of each subset Sis β⁄n × |S| (n - |S|) ≥ β|S|/2 β⁄n units of flow bet. each vtx pair S Why α ≤ O(log n) β:The LP expressing existence of flowis feasible if graph diameter is O(1/β).(uses duality theorem) In a graph with expansion α,diameter is O(log n/α). S (Region growing argument: BFS from S one step at a time; # of edges increases by (1+α) factor each step;reach >1/2 the edges in O(log n/α) steps.)

Approximating expansion via expander flows (A.,Rao, Vazirani 2004) β units of flow originating ateach vtx Route a flow withsome demand graph W= (wij) (wij = flow between i and j) s.t. W is β-regular and has expansion 0.01(“expander flow”) Maximise β. S Easy: α ≥ 0.01 β (Amount of flow leaving each set S is at least 0.01 β |S|.) Main claim: α≤ O(β √log n) Next: Geometry of cuts and how efficiently they can be crossed

Geometry of cuts Cut semimetricdS(i,j) = 1 if i, j on opposite sides of the cut, = 0 else. Sc 1 S (gives embedding into a line) 0 Convex combination of cut semimetrics d(i, j) = ΣS αSdS(i, j) (Gives embedding into l1 : i vi|vi – vj|1= fraction of cuts i, j are across)

Approximating expansion via flows (A.,Rao, Vazirani 2004) β units of flow originating ateach vtx LP formulation: Route a flow with demand graph W= (wij) (wij = flow between i and j) W is β-regular and has expansion 0.01 Maximise β. S Check by computingeigenvalue (“separationoracle”) Main claim: α≤ O(β √log n) Duality Thm Feasibility follows if for every distribution (αS) on balanced cuts, there are Ω(n) disjoint vertex pairs (i1, j1), (i2, j2), … s.t. (a) d(ir, jr) = O(√log n/ α) (b) ir, jrare across Ω(1) fraction of cuts. 1st structure theorem Open: Replace √log n with o(√log n )?(Best lowerbound: log log n [DKSV06])

[ARV04] If G is an α-expander then for every distribution (αS) on balanced cuts, there are Ω(n) disjoint vertex pairs (i1, j1), (i2, j2), … s.t. (a) d(ir, jr) = O(√log n/ α) (b) ir, jr are across Ω(1) fraction of cuts. Thoughts on Structure Thm sink Warmup: If max degree= O(1) and given a single balanced cut, above is true with O(1/α) instead of O(√log n/ α) 1 1 Pf: Max-Flow Min Cut Thm S 4/α 1 α-expansion  Min Cut = Ω(|S|) = Ω(n) =Max-Flow Total capacity = O(n/α)Flow decomposition  Ω(n) flowpathsof length O(1/α) with one endpoint in S and one in Sc 1 (all other edges capacity 4/α ) source

A flow-based O(√log n)-approximation algorithm for expansion For β = 1/n, 2/n, 4/n,… do Try to solve above LP to route a β-regular expander flow in G If succeed, have verified that expansion ≥ 0.01β If fail, use [ARV] technique to find a cut of expansion < O(β√log n)(note: before finding this cut had already verified expansion ≥ 0.01 β/2) (Note: Can be implemented in O(n1.5) time using matrix multiplicativeweight method [Sh09,AK07,AHK05]. Satyen Kale’s talk.)

Suggested research directions Nothing special about routing an expander flow; could use any graph family whose expansion can be verified up to O(1) factor. (Suffices tosolve LP.)Example: Graphs with a few small nonzero eigenvalues (generalizes expanders,which have no small eigenvalues)[A., Barak, Steurer’10] Could also try for o(√log n) approximation in subexponential time. See David Steurer’s talk….

View 2: Use of math programming relaxations Semidefinite Linear Cut semimetricdS(i,j) = 1 if i, j on opposite sides of the cut, = 0 else. Integer program for c-balanced separator (expansion of sets of size ≥ cn) Sc Recall: [LR88]; O(log n)-approximation S [ARV04]: O(√log n) –approximation. (Main obstacle: understanding vectors satisfying triangle inequality condition).

How to round the SDP: 2ndStructure Theorem v1, v2, v3, … : unit vectors in Rn, s.t.avg |vi –vj|2 = Ω(1) (“well-spread”)|vi –vj|2 + |vj-vk|2 ≥ |vi –vk|2 (l22 property; angle subtended by any two pointson the third is nonobtuse; includes l1 as subcase) THM: For Δ = Ω(1/√log n) there exist sets S, T of size Ω(n)s.t.|vi –vj|2 ≥ Δfor all i in S, j in T (S, T are Δ-separated) Δ NB: Implies weak version of 1st Structure Thm: MaxflowMincut applied to S, T yieldsΩ(n) paths of length O(1/α) that cross Ω(1/√log n) fraction of cuts.

O(√log n)-approximation for other cut-like problems • MIN-2-CNF deletion and several graph deletion problems. [Agarwal, Charikar, Makarychev, Makarychev04][’04]. Weighted version of S • MIN-LINEAR ARRANGEMENT [Charikar, Karloff, Rao’05] • General SPARSEST CUT [Chawla-Gupta-Raecke05, A. Lee Naor’06] 0 • Min-ratioVERTEX SEPARATORS and Balanced VERTEX SEPARATORS[Feige, Hajiaghayi, Lee’04] • All Method: SDP rounding using a generalized structure theorem…

Suggested future direction SDP with triangle inequality corresponds to level 2ofLasserre, Lovasz-Schrijver, etc. (see MadhurTulsiani’s talk) Use more powerful SDP relaxations from higher levels? * May need to allow superpolynomial time (rth level  nr time) * Not currently ruled out under UGC.

Cut problems and embeddings General Sparsest Cut: Cost matrix (cij) cij ≥0; Demand matrix (dij) dij ≥ 0; Find SDP relaxation: [LLR94,AR94]: Integrality gap= Minimum distortion incurredwhen embedding l22 metrics intol1(= convex combination of cut semi-metrics)

Geometric Embeddings of Metric Spaces Geometric space, eg l1 Finite metric space (X, d) f f(x) x y d(x,y) f(y) Distortion of f : Minimum Cs.t.d(x, y) ≤ |f(x) –f(y)| ≤ Cd(x,y) [Bourgain’85, LLR94]: Distortion O(log n) into l1, l2 What if X is itself geometric? [Chawla-Gupta-Raecke05, A.-Lee-Naor06]: Distortion O(√log n log log n) for embedding l22 into l1; and embedding l1 into l2

Embedding theorems in one slide Tool 1: Padded decompositions [Krauthgamer,Lee, Mendel,Naor04] Scale S, padding parameter p: Partition probabilistically into piecesof diameter ≤ S, s.t.for all x Pr[x’s partition contains Ball(x, S/p)] ≥ ½ Metric space(X, d) x Line embedding Map each block to 0 with probability ¼; Map x to d(x, zero-block)  0 Tool 2: Use ARV structure theorem to produce padded decompositionsat different scales; combine line embeddings into a single embedding using “measured descent.”

Proving lowerbounds on distortion • [Khot-Vishnoi05] log log n lowerbound; construction inspired by PCPs (hypercontractivityof noisy hypercubes) • [Lee-Naor],[Cheeger,Kleiner,Naor] (log n)εlowerbound; construction based upon Heisenberg group; new notion of differentiation • [Lee-Muharrami] √log n lowerbound; only for embeddingweak l22spaces into l1. Elementary construction and analysis. Open: √log n lowerbound for l22spaces; (log n)εlowerbound for SDP integrality gap of uniform sparsest cut (ie edge expansion).

Algorithm to produce two Δ-separated sets (Δ= 1/√log n) 0.01/√d Easy: Suand Tu are likely to have size Ω(n). Tu Delete any vi in Su and vj in Tus.t. |vi – vj|2 < Δ(repeat until no such pair remains) Su u If Suand Tu still have size Ω(n) output them. Main difficulty: Show that whp only o(n) points get deleted. Obs: Deleted pairs were “stretched”, i.e., |vi – vj|2 < Δ, |<vi – vj, u>| > 0.01/√d Fact: Pr[|<vi – vj, u>| > 0.01 √Δ/√d] = exp(-1/Δ) = exp(-√log n). Too large for union bound

Walks in l22 space |vi –vj|2 + |vj-vk|2 ≥ |vi –vk|2 Δ Δ r steps of squared-length Δ only take youa total squared distance rΔ(i.e., distance √r √Δ) Δ Δ Main proof step: Use measure concentration to prove that for most directionsu there is a walk of length r on stretched edges (v1, v2), (v2, v3),.. (vr, vr+1) so that |<v1 – vr+1, u>| > 0.001 r/√d Pr[such v1, vr+1 exist in the point set] < exp(- r/Δ) < 1/n2

Unique Games Conjecture [Khot03] Given m equations in n variables x1, x2, …, xn of the type axi + b Xj = a (mod 113) s.t.(1-ε) fraction are simultaneously satisfiable, it is NP-hard to satisfy ½ of them simultaneously. Used to prove best inapproximability results for host of problems, includingexpansion problems. Inspired SDP integrality gaps (aka embedding lowerbounds). (See Khot’s talk) (Expansion strikes back) The Achilles heel of UGC appears to be expansion. Better understanding of small-set expansion may disprove UGC. (see Steurer’s talk)

Looking forward to more insight in the next decade! Thank you!

What have we learnt about graph expansion in the new millenium ?