What have we learnt about graph expansion in the new - PowerPoint Presentation

Slide1

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

Sanjeev

Arora

Princeton University &

Center for Computational IntractabilitySlide2

Overview

Last

millenium

: Central role of expansion and expandersRecognizing 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 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 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

C

ayley

graphs of groups, new

algebra-free constructions of optimal expanders, etc.)Slide3

d-regular graph G

d

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|, |

S

c

| >

W

(n))

α

2

/2 ≤

λ

≤ 2α

[

Cheeger

,

Alon

85, Alon-Milman85].

λ

= smallest nonzero

eigenvalue

of

L

aplacian

of G.

Allows us to

recognize

graphs with α =

Ω

(1

)

(

“

expander”) Slide4

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”)Slide5

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 pairWhy 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.)Slide6

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:

G

eometry of cuts

and how efficiently they can be crossedSlide7

Geometry of cuts

S

S

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

01(gives embedding into a line)Convex combination of cut semimetrics d(i, j) = ΣS αS dS

(

i

, j)

(Gives embedding into l

1

:

i

 v

i

|v

i

– v

j

|

1

= fraction of cuts

i

, j are across)Slide8

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 0.01Maximise β.Main claim: α≤ O(β √log n)LP formulation:Duality Thm  Feasibility follows if for every distribution (α

S

) on balanced cuts,

there are

Ω

(n)

disjoint vertex pairs (i

1

, j

1

), (i

2

, j

2

), …

s.t.

(

a

) d(

i

r

,

j

r

) = O(√log n/ α)

(b) i

r, jr

are across Ω(1) fraction of cuts.

Check by computing

eigenvalue (“separation

oracle”)

Open: Replace √log n with o(√log n )

? (Best lowerbound: log log n [DKSV06])

1st structure theoremSlide9

[ARV04] If G is an α-expander then for every distribution (α

S

) 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

1

1

source

1

1

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

S

c

Thoughts on

Structure

ThmSlide10

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 ≥ 0.01βIf 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.) Slide11

S

uggested 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….Slide12

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)-approximationSemidefinite

[ARV04]: O(√log n)

–approximation.

(Main obstacle: understanding vectors satisfying triangle inequality condition).Slide13

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

(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.

|v

i

–v

j

|

2

≥

Δ

for all

i

in S, j in T

(S, T are

Δ-separated

)

Δ

NB: Implies

weak

version of 1

st

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.Slide14

Rounding the SDP

S

T

S, T: Δ-separated sets of size Ω(n)

Do BFS wrt

distance function

d(

i

,j

) = |v

i

–v

j

|

2

,

s

tarting from S and going until you hit T

Output the level of the BFS tree with least expansion.

S

v

i

v

j

d

(S,

i

)

d

(S,

j

)

d

(S, j) – d(S,

i

) ≤ |v

i

–v

j

|

2

Edge (

i

,j

) contributes to cut for ≤ |v

i –vj

|2

levels,and each level cuts

at least |E(O, Oc)| edges.

Claim: This gives a balanced cut (O,

O

c) s.t. |E(O,

Oc)| ≤ SDP

OPT /Δ = O(√log n) SDP

OPTSlide15

O(√log n)-approximation for

other

cut-like

problemsMIN-2-CNF deletion and several graph deletion problems. [

Agarwal, Charikar, Makarychev, Makarychev04][’04]. Weighted version of SMIN-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…Slide16

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.Slide17

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 incurred

when embedding l22 metrics intol1 (= convex combination of cut semi-metrics)Slide18

Finite metric space (X, d)

x

y

d(x,y)Geometric space, eg l1f(x)

f(y)fDistortion 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 l

1

, l

2

[Chawla-Gupta-Raecke05, A.-Lee-Naor06]: Distortion

O(√log n



log log n

)

for embedding l

2

2

into l

1

; and embedding l

1

into l

2

Geometric

Embeddings

of Metric Spaces

What if X is itself geometric?Slide19

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 pieces

of diameter ≤

S

,

s.t.

for all x

Pr

[x’s partition contains

Ball

(x,

S

/

p

)] ≥ ½

x

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 decompositions

at different scales;

combine line

embeddings

into a single embedding using

“measured descent.” Slide20

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 embedding

weak l22 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).Slide21

Proof of Structure Theorems

Recall:

v

1, 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 l

1

metrics, which are a subcase of l

2

2

)

For

Δ

=

Ω

(1/√log n)

there exist sets S, T of size

Ω

(n)

s.t.

|v

i

–v

j

|

2

≥

Δ

for all i in S, j in T (S, T are

Δ-separated)Slide22

Algorithm to produce two Δ

-separated sets (

Δ

= 1/√log n)0.01/√dEasy: Su and

Tu are likely to have size Ω(n).SuTuuDelete any vi in Su and vj in Tu s.t. |vi – v

j

|

2

<

Δ

(repeat until no such pair remains)

If

S

u

and

T

u

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

, |<v

i – vj, u>| > 0.01/√

d Fact:

Pr[|<vi – vj, u>|

> 0.01 √Δ/√d] = exp(-1/Δ

) = exp(-√log n). Too large for union boundSlide23

Walks in l2

2

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 – v

r+1

, u>| > 0.001 r/√d

Pr

[such

v

1

, v

r

+

1

exist in the point set] <

exp

(- r/

Δ

) < 1/n

2

Δ

Δ

Δ

ΔSlide24

Unique Games Conjecture

[Khot03] Given m equations in n variables x

1

, 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, including

expansion 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)Slide25

Looking forward to more insight in the next decade!

Thank you!