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Geometry and Expansion: A survey of recent results

Sanjeev Arora Princeton

( touches upon:

S. A., Satish Rao, Umesh Vazirani, STOC’04;

S. A., Elad Hazan, and Satyen Kale, FOCS’04;

S. A., James Lee, and Assaf Naor, STOC’05

+ papers that are not mine)

| E(S, Sc)|

(G) = min

S

S

S µ V

|S|

|S| < |V|/2

| E(S, Sc)|

c(G) = min

S µ V

|S|

c |V| < |S| < |V|/2

Sparsest Cut / Edge ExpansionG = (V, E)

c- balanced separator

Both NP-hard

Why these problems are important

- Analysis of random walks, PRAM simulation, packet routing, clustering, VLSI layout etc.
- Underlie many divide-and-conquer graph algorithms (surveyed by Shmoys’95)
- Discrete analog of isoperimetry; useful in Riemannian geometry (via 2nd eigenvalue of Laplacian (Cheeger’70)
- Graph-theoretic parameters of inherent interest (cf. Lipton-Tarjan planar separator theorem)

The three main characters

Expansion

Isoperimetry

(continuous analog of expansion)

Geometry

(and geometric embeddings of finite metric spaces)

Outcome: New plog n –approximations for various NP-hard problems;

Derived using geometric insights, & which led to new geometry thms.

3) Embeddings of finite metric spaces into l1

- Geometric approach; more general result (but still O(log n) approximation)

- Eigenvalue approaches (Cheeger’70, Alon’85, Alon-Milman’85)Only yield factor n approximation. 2c(G) ¸ (G) ¸ c(G)2 /2

2) O(log n) -approximationvia LP (multicommodity flows)

(Leighton-Rao’88)

- Approximate max-flow mincut theorems
- Region-growing argument

(Linial, London, Rabinovich’94, AR’94)

log n

New results of [ARV’04]- O( ) -approximation to sparsest cut and conductance
- O( )-pseudoapproximation to c-balanced separator (algorithm outputs a c’-balanced separator, c’ < c)
- Existence of expander flows in every graph (approximate certificates of expansion)

Disparate approaches from previous slide get “unified”

- Outline:
- Graph partitioning problems: intro and history
- New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani]
- Uses of “S. T.” in geometric embeddings
- Introduction to expander flows and O(n2) time algorithms
- Outline of proof of “S. T.”
- Open problems

Next: Semidefinite relaxations for c-balanced separator

(and how to round the solution)

| E(S, Sc)|

c(G) = min

S µ V

|S|

c |V| < |S| < |V|/2

Semidefinite relaxation for

c-balanced separator|vi –vj|2/4 =1

|vi –vj|2 =0

+1

S

-1

“cut semimetric”

Find unit vectors

in <n

Assign {+1, -1} to v1, v2, …, vn to minimize

(i, j) 2 E |vi –vj|2/4

Subject to i < j |vi –vj|2/4 ¸ c(1-c)n2

Triangle inequality

|vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k

Vi

Vj

Vk

Unit l22 spaceUnit vectors v1, v2,… vn2<d

|vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k

non obtuse !

Example: Hypercube {-1, 1}k

|u – v|2 = i |ui – vi|2

= 2 i |ui – vi| = 2 |u – v|1

In fact, l2 and l1 are subcases of l22

1

p log n

Structure Theorem for l22 spaces [ARV’04]Subsets S and T are -separated if

for every vi2 S, vj2 T |vi –vj|2¸

<d

G = Graph in which (i,j) is an edge

iff |vi –vj|2·

¸

Thm: If i< j |vi –vj|2 = (n2) then 9S, T of size (n)

that are -separated for = ( 1 )

Equiv: G is an “expander” )·

) |E(R, Rc)| · SDPopt /

· O( SDPopt)

Main thm ) O( )-approximationlog n

v1, v2,…, vn2<d is optimum SDP soln;

SDPopt = (i, j) 2 E |vi –vj|2

S, T : –separated sets of size (n)

Do BFS from S until you hit T. Take the level

of the BFS tree with the fewest edges and

output the cut (R, Rc) defined by this level

(i, j) 2 E |vi –vj|2¸ |E(R, Rc)| £

Other new -approximation algorithms

- MIN-2-CNF deletion and several graph deletion problems. [Agarwal, Charikar, Makarychev, Makarychev’05]
- MIN-LINEAR ARRANGEMENT [Charikar, Karloff, Rao’05]
- General SPARSEST CUT [A., Lee, Naor ’05]
- Min-ratioVERTEX SEPARATORS and Balanced VERTEX SEPARATORS[ Feige, Hajiaghayi, Lee, ’05]

Example: Structure Theorem (Agarwal, Charikar, Makarychev2 ‘05)

d : directed version of l22 metric;

w: weight function on the nodes

G = (V, E): any graph on the nodes.

S

There exists a subset S that contains 1/10 of the total weight and such that e leaves S d(e) is at

Most p log n £e 2 E d(e).

All use the Structure Theorem (+ other ideas)

(Useful in rounding SDP for MIN-2CNF-DELETION.)

- Outline:
- Graph partitioning problems: intro and history
- New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani]
- Geometric embeddings of metric spaces
- Introduction to expander flows and O(n2) time algorithms
- Outline of proof of “S. T.”
- Open problems

<k(with l2 norm)

Finite metric space (X, d)

f(x)

y

f

d(x,y)

x

f(y)

distortion of f is minimum C>1 such that

d( x, y) · |f(x ) – f( y)|2·C d( x, y) 8 x, y

Thm (Bourgain’85): For every n-point metric space, a map exists with distortion O(log n)

[LLR’94]: Efficient algorithm to find the map; Proof that O(log n) cannot be improved in general

Qs: Improve O(log n) for X = l22 (say) or l1 ?

Embeddings and Cuts (LLR’94, AR’94)

Recall: Cut semi-metric

Fact: Metric (X, d) embeds isometricallyin l1 iff it can be written as a positive combination of cut semimetrics

1

0

Embedding l22 into l1 gives a way to produce cuts from SDP solution

Status report of this area

Best upperbound

Best lowerbound

Disproves Goemans-Linial conjecture

log n

[Bourgain’85]

Uses fourier techniques developed for PCPs!

log0.75 n

[Chawla,Gupta,Racke ’04]

Exactly the integrality

gap of SDP for general

SPARSEST CUT

[LLR’94, AR’94]

log0.5 n log log n

[A., Lee, Naor’04]

Note: l2µ l1µ l22

Ai

Embedding Upperbounds:Frechet’s recipe to embed metric space (X, d) into RkPick k suitable subsets A1, A2, …, Ak of X

Map x 2 X to (d(x, A1), d(x, A2), … , d(x, Ak))

Note: d(x, A1) – d(y, A1) · d(x, y)

Why S.T. useful: If Sobtained from S.T., then in the mapping x ! d(x, S), “many” x’s (namely, all those in T) map far from 0.

In recent embeddings, Ai’s are chosen using S.T.and “Measured descent” idea of [Krauthgamer, Lee, Naor, and Mendel’04]

Embedding lowerbounds (Khot-Vishnoi’05)

Explicit unit- l22 space (X, d) that requires distortion log log log n into l1

Main observation: Need good handle on cut structure of X

Use hypercube as building block !

Cut

´ Boolean Function

Number of cut edges = average sensitivity

(Fourier analysis a la KKL, Friedgut, Hastad,

Bourgain etc. ) isoperimetric theorems)

[Khot-Naor]: Lowerbounds for embedding earth-mover & edit metrics into l1

- Outline:
- Graph partitioning problems: intro and history
- New approximation algorithm via semidefinite programming (+ analysis using “Structure Theorem”) [A., Rao, Vazirani]
- Outline of proof of “S. T.”
- Uses of “S. T.” in geometric embeddings
- Introduction to expander flows and O(n2) time algorithms
- Open problems

S

Our Thm: If G has expansion , then a D-regular expander flow exists in it where D=

Expander flows: Motivation“Expander”

G = (V, E)

Idea: Embed a D-regular (weighted) graph such that 8 S w(S, Sc) = (D |S|)

(*)

S

(certifies expansion = (D) )

Weighted Graph w satisfies (*) iff L(w) = (1) [Cheeger]

Cf. Jerrum-Sinclair, Leighton-Rao(embed a complete graph)

Can be found in O(n2) time (A., Hazan, Kale ’04)

Example of expander flow

n-cycle

Take any 3-regular expander on n nodes

Put a weight of 1/3n on each edge

Embed this into the n-cycle

Routing of edges does not exceed any capacity ) expansion =(1/n)

S

- Outline:
- Graph partitioning problems: intro and history
- Uses of “S. T.” in geometric embeddings
- Introduction to expander flows and O(n2) time algorithms
- Outline of proof of “S. T.”
- Open problems

Outline of proof of S. T.

(Algorithm to produce -separatedsets S, T, of size (n) )

d

“Stretched pair”: vi, vj such that |vi –vj|2· and

| h vi –vj, u i | ¸ 0.01

d

Algorithm to produce two –separated sets<d

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

u

Tu

Delete any vi2 Su, vj2 Tu s.t. |vi –vj|2 < . (till no such pair remains)

Su

If Su, Tu still have size (n), output them

Main difficulty: Show that whp only o(n) points get deleted

Obs: Deleted pairs are stretched and they form a matching.

-t2

/2

e

1

1

d

d

= O( 1 )

Stretched pair: |vi –vj|2 < ; |<vi –vj, u>| > 0.01

d

Naïve analysis of random projection failsv

<d

u

<u, v> ??

standard deviations

E[# of stretched pairs] = n2 exp(-) À n

Vi

0.01

d

Proof by contradiction: Suppose matching of (n) size exists with probability (1)…

….stretched pairs are almost everywhere you look!

Vj

u

Ball (vi , )

Idea: Put stretched pairs together; derive very improbable event

Vi

Vj

Vk

s

s

s

s

Walks in unit l22 spaceUnit vectors v1, v2,… vn2<d

|vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k

Angles are non obtuse

Taking r steps of length s

only takes you squared distance rs2

(i.e. distance r s)

squared distance rs2 (distance r s)

s

s

0.01

<vfinal –v0, u> ¸ r

0.01

0.01

d

d

d

s

s

r

Projection =

£ standard deviation

Proof by contradiction (contd.)Claim: 9walk on stretched edges

VERY UNLIKELY IF r large enough) Walk impossible (CONTRADICTION)

Stretched pair: |vi –vj|2 < ; |<vi –vj, u> ¸ 0.01

d

….

u

How to produce walk: delicate argument; measure concentration

|vfinal –v0|· r

OPEN PROBLEMS

- Better approximation factor than O( )? (For general SPARSEST CUT, log log n “lowerbound” )
- Better distortion bound for embedding l22 into l1?( upperbound v/s loglog n lowerbound.)
- Remove need for solving SDPs (i.e., design combinatorial algorithms) (similar to one for SPARSEST CUT from [A., Hazan, Kale] )
- O(m) time algorithm for SPARSEST CUT instead of O(n2)? (not known even for Leighton-Rao’88 O(log n) approximation)
- Other applications of expander flows? (Useful in some geometric results [Naor, Rabani, Sinclair’04])

Looking forward to more progress…

Thanks !

(D) ·(G) ·O(D )

log n

log n

New Result (A., Hazan, Kale;FOCS’04)O(n2) time algorithm that given any graph G finds for some D >0

- a D-regular expander flow
- a cut of expansion O( D )

Ingredients: Approximate eigenvalue computations; Approximate flow computations (Garg-Konemann; Fleischer)

Random sampling (Benczur-Karger + some more)

Idea: Define a zero-sum game whose optimum solution is an expander

flow; solve approximately using Freund-Schapire approximate solver.

Expander flows: LP view

· 1

LP feasible )¸(D)

· D

Thm [ARV]:90 s.t. the LP is

feasible with D = /√log n

G

Open problems (circa April’04)

O(n2) time;

[A., Hazan, Kale]

- Better running time/combinatorial algorithm?
- Improve approximation ratio to O(1); better rounding??(our conjectures may be useful…)
- Extend result to other expansion-like problems (multicut, general sparsest cut; MIN-2CNF deletion)
- Resolve conjecture about embeddability of l22 into l1; of l1 into l2
- Any applications of expander flows?

Integrality gap is (log n) [Charikar]

log3/4 n distortion; [Chawla,Gupta, Racke]

Yes [Naor,Sinclair,Rabani]

Better embeddings of lp into lq [Lee]

Various new results

O(n2) time combinatorial algorithm for sparsest cut

(does not use semidefinite programs)

[A., Hazan, Kale’04]

New results about embeddings: (i) lp into lq[J. Lee’04]

(ii) l22 and l1 into l2[CGR’04]

(approx for general sparsest cut)

Clearer explanation of expander flows and their connection to

embeddings [NRS’04]

Formal statement : 90 >0 s.t. foll. LP is feasible for d = (G)

Pij = paths whose endpoints are i, j

8i jp 2 Pij fp = d (degree)

8e 2 E p 3 e fp· 1 (capacity)

8S µ V i 2 S j 2 Scp 2 Pij fp¸0 d |S| (demand graph is an expander)

fp¸ 0 8 paths p in G

A concrete conjecture (prove or refute)

G = (V, E); = (G)

For every distribution on n/3 –balanced cuts {zS} (i.e., SzS =1)

there exist (n) disjoint pairs (i1, j1), (i2, j2), ….. such that for each k,

- distance between ik, jk in G is O(1/ )

- ik, jk are across (1) fraction of cuts in {zS} (i.e., S: i 2 S, j 2 Sc zS = (1) )

Conjecture ) existence of d-regular expander flows for d =

log n

Example of l22 space: hypercube {-1, 1}k

|u – v|2 = i |ui – vi|2

= 2 i |ui – vi| = 2 |u – v|1

In fact, every l1 space is also l22

Conjecture (Goemans, Linial): Every l22 space is l1 up to

distortion O(1)

0

0

1

1

Semidefinite

LP Relaxations for c-balanced separatorMin (i, j) 2 E Xij

0 · Xij· 1

Motivation: Every cut (S, Sc) defines a (semi) metric

Xij2 {0,1}

Xij + Xj k¸ Xik

i< j Xij¸ c(1-c)n2

There exist unit vectors v1, v2, …, vn2<n

such that Xij = |vi - vj|2 /4

Semidefinite relaxation (contd)

Min (i, j) 2 E |vi –vj|2/4

|vi|2 = 1

|vi –vj|2 + |vj –vk|2¸ |vi –vk|28 i, j, k

i < j |vi –vj|2¸ 4c(1-c)n2

Unit l22 space

Many other NP-hard problems have similar relaxations.

If any vi2 Su and vj2 Tu satisfy

|vi –vj|2·,

delete them and

repeat until no such vi, vj remain

0.01

d

“Stretched pair”: vi, vj such that |vi –vj|2· and

| h vi –vj, u i | ¸ 0.01

d

Algorithm to produce two –separated sets<d

Check if Su and Tu have size (n)

u

Tu

Su

If Su, Tu still have size (n), output them

Main difficulty: Show that whp only o(n) points get deleted

Obs: Deleted pairs are stretched and they form a matching.

S

Next 10-12 min: Proof-sketch of Structure Thm

( algorithm to produce -separated S, T of size (n);

= 1/ )

“Stretched pair”: vi, vj such that |vi –vj|2· and

| h vi –vj, u i | ¸ 0.01

d

O( 1 ) £ standard deviation

= exp( - )

log n

) PrU [ vi, vj get stretched] = exp( - 1 )

E[# of stretched pairs] = O( n2 ) £ exp(- )

logn

“Matching is of size o(n) whp” : naive argument fails|vfinal - vi| < r

= O( r ) x standard dev.

| <vfinal – vi, u>| ¸ 0.01r

0.01

0.01

0.01

r

d

d

d

d

Generating a contradiction: the walk on stretched pairsContradiction if r is large enough!

Vj

vfinal

Vi

r steps

u

Reason: Isoperimetric inequality for

spheres

Measure concentration (P. Levy, Gromov etc.)<d

A : measurable set with (A) ¸ 1/4

A : points with distance · to A

A

(A) ¸ 1 – exp(-2 d)

A

(approximate certificates of expansion)

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