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

Sanjeev AroraPrinceton University &

Center for Computational Intractability

OverviewLast millenium:

• Central role of expansion and expanders• Recognizing expander graphs via eigenvalues (Cheeger71,Alon-Milman85)• O(log n)-approximation via flows (Leighton-Rao88); region-growing technique;• Close connection to metric embeddings; O(log n) approximation for general

sparsest 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 SDP and 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)• Lowerbounds for 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.)

d-regular graph Gd

vertex set S

Graph Expansion

expansion(S) = # edges leaving S

d |S|

Important concept: derandomization, network routing, coding theory,Markov chains, differential geometry, group theory

a-expander: expansion(S) ≥ a for all S (co-NP-hard to recognize)

(often will restrict attention wlog to “balanced” sets: |S|, |Sc| > W(n))

α2/2 ≤ λ ≤ 2α [Cheeger, Alon 85, Alon-Milman85]. λ= smallest nonzero eigenvalue of Laplacian of G.Allows us to recognize graphs with α = Ω(1) (“expander”)

Approximating expansion via flows

[Leighton-Rao’88] O(log n)-approximation.Find largest b s.t. we can simultaneously route b/n units offlow between every vertex pair.(“embed a complete graph”)

S

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

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)

S

β units of flow originating ateach vtx

Route a flow with some 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.

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

S

S cCut semimetric

dS(i,j) = 1 if i, j on opposite sides of the cut, = 0 else.

0

1(gives embedding into a line)

Convex combination of cut semimetrics

d(i, j) = ΣS αS dS(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)

S

β units of flow originating ateach vtx

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.

Main claim: α≤ O(β √log n)

LP formulation:

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, jr are across Ω(1) fraction of cuts.

Check by computingeigenvalue (“separationoracle”)

Open: Replace √log n with o(√log n )? (Best lowerbound: log log n [DKSV06])

1st structure theorem

[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.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)

S Pf: Max-Flow Min Cut Thm

11

source

11

sink

(all other edges capacity 4/α )

4/α

α-expansion Min Cut = Ω(|S|) = Ω(n) =Max-Flow

Total capacity = O(n/α)

Flow decomposition Ω(n) flowpaths of length O(1/α) with one endpoint in S and one in Sc

Thoughts on Structure Thm

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

S

S cCut semimetric

dS(i,j) = 1 if i, j on opposite sides of the cut, = 0 else.

Recall: Integer program for c-balanced separator (expansion of sets of size ≥ cn)

Linear

[LR88]; O(log n)-approximation

Semidefinite

[ARV04]: O(√log n) –approximation.(Main obstacle: understanding vectors satisfying triangle inequality condition).

How to round the SDP: 2nd Structure 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

(l2

2 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 ≥ Δ for all i in S, j in T

(S, T are Δ-separated)Δ

NB: Implies weak version of 1st Structure Thm: Maxflow Mincut applied to S, T yields Ω(n) paths of length O(1/α) that cross Ω(1/√log n) fraction of cuts.

Rounding the SDP

S

TS, T: Δ-separated sets of size Ω(n)

Do BFS wrt distance function d(i,j) = |vi –vj|2 ,starting from S and going until you hit T

Output the level of the BFS tree with least expansion.

S

vi

vj

d(S, i)d(S, j)

d(S, j) – d(S, i) ≤ |vi –vj|2

Edge (i,j) contributes to cut for ≤ |vi –vj|2 levels,and each level cuts at least |E(O, Oc)| edges.

Claim: This gives a balanced cut (O, Oc) s.t. |E(O, Oc)| ≤ SDPOPT /Δ = O(√log n) SDPOPT

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-ratio VERTEX 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 2 of Lasserre, Lovasz-Schrijver, etc. (see Madhur Tulsiani’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 l2

2 metrics intol1 (= convex combination

of cut semi-metrics)

Finite metric space (X, d)

xyd(x,y)

Geometric space, eg l1

f(x)

f(y)

f

Distortion of f : Minimum C s.t. d(x, y) ≤ |f(x) –f(y)| ≤ C d(x,y)

[Bourgain’85, LLR94]: Distortion O(log n) into l1, l2

[Chawla-Gupta-Raecke05, A.-Lee-Naor06]: Distortion O(√log n log log n) for embedding l2

2 into l1; and embedding l1 into l2

Geometric Embeddings of Metric Spaces

What if X is itself geometric?

Embedding theorems in one slide

Tool 1: Padded decompositions [Krauthgamer,Lee, Mendel,Naor04]

Metric space(X, d)

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)] ≥ ½x

Line embedding

0 Map each block to 0 with probability ¼;

Map x to d(x, zero-block)

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(hypercontractivity of noisy hypercubes)

• [Lee-Naor],[Cheeger,Kleiner,Naor] (log n)ε lowerbound; construction based upon Heisenberg group; new notion of differentiation

• [Lee-Muharrami] √log n lowerbound; only for embeddingweak l2

2 spaces into l1. Elementary construction and analysis.

Open: √log n lowerbound for l22 spaces;

(log n)ε lowerbound for SDP integrality gap of uniform sparsest cut(ie edge expansion).

Proof of Structure TheoremsRecall:

v1, v2, v3, … : unit vectors in Rd, s.t.avg |vi –vj|2 = Ω(1) (“well-spread”)

|vi –vj|2 + |vj-vk|2 ≥ |vi –vk|2

Δ

(Recall: 1st Structure Theorem concerned distributions of cuts,which correspond to l1 metrics, which are a subcase of l2

2)

For Δ = Ω(1/√log n) there exist sets S, T of size Ω(n) s.t.

|vi –vj|2 ≥ Δ for all i in S, j in T

(S, T are Δ-separated)

Algorithm to produce two Δ-separated sets (Δ= 1/√log n)0.01/√d

Easy: Su and Tu are likely to have size Ω(n).

Su

Tu

u

Delete any vi in Su and vj in Tu s.t. |vi – vj|2 < Δ(repeat until no such pair remains)

If Su and 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

r steps of squared-length Δ only take youa total squared distance rΔ (i.e., distance √r √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

ΔΔ

Δ Δ

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!