Sparsest Cut

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# Sparsest Cut - PowerPoint PPT Presentation

S.  ( G) = min |E(S, S)| |S|.  ( G) = min |E(S, S)| |S|. S µ V. S µ V. c. |S| ¸ c ¢ |V|. Sparsest Cut. G = (V, E). S. c- balanced separator. Both NP-hard. Why these problems are important.

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S

(G) = min |E(S, S)|

|S|

 (G) = min |E(S, S)|

|S|

S µ V

S µ V

c

|S| ¸ c ¢ |V|

Sparsest Cut

G = (V, E)

S

c- balanced separator

Both NP-hard

Why these problems are important
• Arise in analysis of random walks, PRAM simulation, packet routing, clustering, VLSI layout etc.
• Underlie many divide-and-conquer graph algorithms (surveyed by Shmoys’95)
• Related to curvature of Riemannian manifolds and 2nd eigenvalue of Laplacian (Cheeger’70)
• Graph-theoretic parameters of inherent interest (cf. Lipton-Tarjan planar separator theorem)

3) Embeddings of finite metric spaces into l1

(Linial, London, Rabinovich’94)

Previous approximation algorithms
• Eigenvalue approaches (Cheeger’70, Alon’85, Alon-Milman’85)2c(G) ¸L(G) ¸ c(G)2/2 c(G) = minS µ V E(S, Sc)/ E(S)

2) O(log n) -approximation via multicommodity flows (Leighton-Rao 1988)

• Approximate max-flow mincut theorems
• Region-growing argument
• Geometric approach; more general result

log n

log n

Our results
• 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)

1

0

0

1

1

Semidefinite

LP Relaxations for c-balanced separator

Min (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

Vi

Vj

Vk

s

s

s

s

l22 space

Unit 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)

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)

log n

Our Main Theorem

Two subsets S and T are -separated if

for every vi2 S, vj2 T |vi –vj|2¸

<d

¸ 

Thm: If i< j |vi –vj|2 = (n2) then there exist two sets S, T

of size (n) that are  -separated for  = ( 1 )

log n

log n

) |E(R, Rc)| · SDPopt /

· O( SDPopt)

Main thm ) O( )-approximation

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)| £

-t2

/2

e

log n

1

1

d

d

Pru[ projection exceeds 2 ] < 1/n2

d

Projection onto a random line

v

<d

u

<u, v> ??

If any vi2 Su and vj2 Tu satisfy

|vi –vj|2·,

delete them

and repeat until no such vi, vj can be found

0.01

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

| h vi –vj, u i | ¸ 0.01

d

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.

| h vi –vj, u i | ¸ 0.01

d

log n

O( 1 ) £ standard deviation

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

= exp( - )

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

logn

“Matching is of size o(n) whp” : trivial argument fails

Vi

0.01

d

Suppose with probability (1) there is a matching of (n) stretched pairs

Vj

u

Ball (vi , )

|vfinal - vi| < r 

| <vfinal – vi, u>| ¸ 0.01r

0.01

0.01

0.01

r

d

d

d

d

= O( r ) x standard dev.

The walk on stretched pairs

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

A : points with distance · to A

(A) ¸ 1 – exp(-2 d)

A

log n

S

Our Thm: If G has expansion , then a d-regular expander flow can be routed 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|)

(*)

(certifies expansion = (d) )

S

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

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

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)

log n

Formal statement : 90 >0 such that following LP is feasible for d = (G)

Pij = paths whose endpoints are i, j

8i jp 2 Pij fp = d (degree)

8e 2 E p 3 e fp· 1 (capacity)

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

fp¸ 0 8 paths p in G

)

(d) ·(G) · O(d )

log n

log n

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.

Open problems
• 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
• Any applications of expander flows?
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 = 