Geometry and Expansion: A survey of some
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 & JAMS’08
S.A., S. Kale STOC 2007.
+ papers that are not mine)
Outline:
• Graph partitioning problems: intro and history
•New approximation via expander flows.
• New approximation algorithm via semidefinite programming
(+ analysis using “Structure Theorem”)
• Outline of proof of “S. T.”
• Uses of “S. T.” in geometric embeddings
• Open problems
[A., Rao, Vazirani]
Sparsest Cut / Edge Expansion
G = (V, E)
S
(G) = min
| E(S, Sc)|
SµV
S
|S| < |V|/2
c- balanced separator
c(G) = min
Both NP-hard
| E(S, Sc)|
SµV
c |V| < |S| < |V|/2
|S|
|S|
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)
Previous approximation algorithms
1) 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) -approximation via LP (multicommodity flows)
(Leighton-Rao’88)
• Approximate max-flow mincut theorems
• Region-growing argument
3) Embeddings of finite metric spaces into l1
(Linial, London, Rabinovich’94,
• Geometric approach; more general result
AR’94)
(but still O(log n) approximation)
New results of [ARV’04]
1. O( log n ) -approximation to sparsest cut and conductance
2. O( log n )-pseudoapproximation to c-balanced separator
(algorithm outputs a c’-balanced separator, c’ < c)
3. Existence of expander flows in every graph
(approximate certificates of expansion)
Disparate approaches from previous slide get “unified”
Subsequent work: [AHK’05],[AK’07],[S’09]: O(m + n1.5 ) time!
The three main characters
Expansion
Isoperimetry
(continuous analog of expansion)
Geometry
(and geometric embeddings of finite metric spaces)
Identifying sparse cuts via “traffic flows”
Approach 1: traffic congestion identifies
sparse cuts
[SM’87]: Stress a network by
passing traffic “flow” through
it. Look at congested edges
to identify sparse cuts
[LR88] O(log n) approximation
to sparsest cut. Route 1 unit of
traffic between every
pair of nodes
[ARV’04] Traffic flow is like embedding a weighted graph.
wij = amount of traffic from i to j
Solve a math program to find the “right” flow pattern
([AHK’05] Do it in O(n2) time)
Expander traffic flows
G = (V, E)
S
[ARV’04]
A D-regular flow graph s.t.
S
8 S w(S, Sc) = ( D |S|)
(*)
(certifies expansion = (D) )
Weighted Graph w satisfies (*)
iff
L(w) = (1) [Cheeger]
Our Thm: If G has expansion , then a D-regular expander flow
exists in it where D=
Formal statement : 9 0 >0 s.t. foll. LP is feasible for D = (G)
log n
Pij = paths whose endpoints are i, j
8i
j p 2 Pij fp = D
8e 2 E p 3 e fp · 1
8S µ V i 2 S j 2 Sc p 2 Pij fp ¸ 0 D |S|
fp ¸ 0
8 paths p in G
WHY IS THIS FEASIBLE???
(degree)
(capacity)
(demand graph is
an expander)
Feasibility Criterion for LP on prev. slide
(via Farkas’s Lemma)
Existence of such i, j proved in [ARV’04]. When fail to
find such i, j, we find a cut of small expansion
Overall approximation algorithm via flows
Try to solve above LP to find D-regular expander flow
If succeed, have verified that expansion is > D/10.
If fail, then use [ARV04] ideas to find a cut of capacity
Note: Before finding this cut already had D/2-regular flow
Next: The SDP-based approach to
Graph partitioning (ARV’04)
Semidefinite relaxation for
c-balanced separator
|vi –vj|2/4 =1
|vi –vj|2 =0
S
+1
c(G) = min
S
| E(S, Sc)|
SµV
-1
|S|
c |V| < |S| < |V|/2
Find unit vectors
in <n
semimetric” Assign {+1, -1} to v1, v2, …, vn to minimize
“cut
(i, j) 2 E |vi –vj|2/4
Triangle
inequality
Subject to i < j |vi –vj|2/4 ¸ c(1-c)n2
|vi –vj|2 + |vj –vk|2 ¸ |vi –vk|2 8 i, j, k
Unit l22 space
Unit vectors v1, v2,… vn 2 <d
Vi
Vj
|vi –vj|2 + |vj –vk|2 ¸ |vi –vk|2
8 i, j, k
non obtuse !
Vk
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
Structure Theorem for l22 spaces
<d
[ARV’04]
Subsets S and T are -separated if
for every vi 2 S, vj 2 T |vi –vj|2 ¸ 
¸
Thm: If
i< j |vi –vj|2 = (n2) then 9 S, T of size (n)
that are  -separated for  = ( 1 )
log n
Main thm ) O(
log n)-approximation
v1, v2,…, vn 2 <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

d(S, i) d(S, j)
j
S
i
(i, j) 2 E |vi –vj|2
¸
|E(R, Rc)| £ 
) |E(R, Rc)| · SDPopt /
· O(
log nSDPopt)
Other new
log n
-approximation algorithms
• MIN-2-CNF deletion and several graph deletion problems. [Agarwal,
Charikar, Makarychev, Makarychev’04]. Weighted version of S.T.
• MIN-LINEAR ARRANGEMENT [Charikar, Karloff, Rao’04]
• General SPARSEST CUT [A., Lee, Naor ’04]
•
Min-ratio VERTEX SEPARATORS and Balanced VERTEX
SEPARATORS [ Feige, Hajiaghayi, Lee, ’04]
All use the Structure Theorem (+ other ideas)
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
(Algorithm to produce  -separated
T
sets S, T, of size (n) )
• Introduction to expander flows and O(n2) time algorithms
S
• Open problems
Algorithm to produce two  –separated sets
<d
u
Tu
Easy: Su and Tu likely to have size (n)
Delete any vi 2 Su, vj 2 Tu s.t.
|vi –vj|2 < . (till no such pair remains)
Su
0.01
If Su, Tu still have size (n), output them
d
Main difficulty: Show that whp only o(n) points get deleted
“Stretched pair”: vi, vj such that |vi –vj|2 ·  and
| h vi –vj, u i | ¸ 0.01
d
Obs: Deleted pairs are stretched and they form a matching.
Naïve analysis of random projection fails
<d
v
e
u
<u, v> ??
Stretched pair: |vi –vj|2 < ;
1
1
d
d
|<vi –vj, u>| > 0.01
2
-t /2
= O( 1 )

standard deviations
d
E[# of stretched pairs] = n2 exp(-) À n
Proof by contradiction: Suppose matching of (n) size exists
with probability (1)…
….stretched pairs are almost everywhere you look!
Vj
Ball (vi , )
u
Vi
0.01
d
Idea: Put stretched pairs together; derive very improbable event
Walks in unit l22 space
Unit vectors v1, v2,… vn 2 <d
Vi
Vj
|vi –vj|2 + |vj –vk|2 ¸ |vi –vk|2
8 i, j, k
Vk
Angles are non obtuse
s
s
s
s
Taking r steps of length s
only takes you squared distance rs2
(i.e. distance r s)
Proof by contradiction
Claim: 9walk on stretched edges
(contd.)
s
s
r £ standard deviation
Projection =
s
s
r steps of length s )

VERY UNLIKELY
IF distance
r large enough
squared
rs2 (distance
) Walk impossible (CONTRADICTION)
r s)
Stretched pair: |vi –vj|2 < ; |<vi –vj, u> ¸ 0.01
d



….
0.01
0.01
<vfinal –v0, u> ¸ r 0.01
u
Why walk dis possible:ddelicate|vargument;
measure dconcentration
final –v0| · r 
Outline:
• Graph partitioning problems: intro and history
• New approximation algorithm via semidefinite programming
(+ analysis using “Structure Theorem”)
• Outline of proof of “S. T.”
• Geometric embeddings of metric spaces
• Open problems
[A., Rao, Vazirani]
<k (with l2 norm)
Finite metric space (X, d)
y
x
f(x)
f
d(x,y)
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]:
EfficientO(log
algorithm
to find
Qs: Improve
n) for
X =the
l22map;
(say) or l1 ?
Proof that O(log n) cannot be improved in general
Embeddings and Cuts (LLR’94, AR’94)
Recall: Cut semi-metric
1
Fact: Metric (X, d) embeds isometrically
in l1 iff it can be written as a positive
combination of cut semimetrics
0
Embedding l22 into l1 gives a way to produce cuts from SDP solution
Status report of this area
Best lowerbound
l1 into l2
l22 into l1
Exactly the integrality
gap of SDP for general
SPARSEST CUT
[LLR’94, AR’94]
l22 into l2
log0.5 n
[Enflo’69]
1.16
[Zatloukal’04]
Superconstant
[Khot,
Vishnoi’04]
(logn)0.01
[Cheeger,Kleine
r, ‘08]
log0.5 n
[Enflo’69]
Best upperbound
Disproves
log n
Goemans-Linial
[Bourgain’85]
conjecture
Uses fourier
log0.75 techniques
n
developed
[Chawla,Gupta,Racke
’04]
for PCPs!
log0.5 n log log n
[A., Lee, Naor’04]
Uses new metric
differentiation
techniques
Upperbounds:
Frechet’s recipe to embed metric space (X, d) into Rk
Pick 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)
In recent embeddings, Ai’s are chosen using
S.T.and “Measured descent” idea of
[Krauthgamer, Lee, Naor, and Mendel’04]
x
Ai
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)
OPEN PROBLEMS
• Better approximation factor than O(
)?
(log log n “lowerbound” assuming UGC )
• Better distortion bound for embedding l22 into l1?.) (
v/s
upperbound
lowerbound
• Combinatorial approximation algorithms for other problems ?
(similar to one for SPARSEST CUT from [A., Hazan, Kale] )
• Other applications of expander flows?
(Useful in some geometric results [Naor, Rabani, Sinclair’04])
• Ways to use spectral ideas a la [ABS’10] for SPARSEST CUT?
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)
Other extensions of flow-based techniques
• Generalization to problems other than sparsest cut [A.,
Kale07] “Primal-dual approach to SDP.”
• Very fast algorithms for O(log n) approximation: O(n1.5 +
m) time (Faster than [LR88] type algorithms!)
• Very simple algorithms; use only maxflow and
eigenvalue computations [KRV06]
Looking forward to more progress…
Thanks !
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 log n )
)
(D) · (G) ·O(D
log )n
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
·D
G
LP feasible )  ¸ (D)
Thm [ARV]: 9 0 s.t. the LP is
feasible with D = /√log n
Open problems (circa April’04)
2) time;
O(n
• Better running time/combinatorial algorithm?
[A., Hazan, Kale]
• 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)
Integrality gap is (log n) [Charikar]
• Resolve conjecture about embeddability of l22 into l1; of
l1 into l2
log3/4 n distortion; [Chawla,Gupta, Racke]
•
Any applications of expander flows?
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]
A concrete conjecture (prove or refute)
G = (V, E);  = (G)
For every distribution on n/3 –balanced cuts {zS} (i.e., S
zS =1)
there exist (n) disjoint pairs (i1, j1), (i2, j2), ….. such that for each k,
•
distance between ik, jk in G is O(1/ )
• i k, j k
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
log n
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)
Semidefinite
LP Relaxations for c-balanced separator
Min (i, j) 2 E Xij
Motivation: Every cut (S, S
0c) ·defines
Xij · 1a (semi) metric
Xij 2 {0,1}
1
0
1
1
0
Xij + Xj k ¸ Xik
 i< j Xij ¸ c(1-c)n2
There exist unit vectors v1, v2, …, vn 2 <n
such that
Xij = |vi - vj|2 /4
Semidefinite relaxation (contd)
Min (i, j) 2 E
Unit l22 space
|vi –vj|2/4
|vi|2 = 1
|vi –vj|2 + |vj –vk|2 ¸ |vi –vk|2
8 i, j, k
i < j |vi –vj|2 ¸ 4c(1-c)n2
Many other NP-hard problems have similar relaxations.
Algorithm to produce two  –separated sets
<d
u
Tu
Check if Su and Tu have size (n)
If any vi 2 Su and vj 2 Tu satisfy
|vi –vj|2 · ,
delete them and
Su
0.01
d
repeat until no such vi, vj remain
If Su, Tu still have size (n), output them
Main difficulty: Show that whp only o(n)2 points get deleted
“Stretched pair”: vi, vj such that |vi –vj| ·  and
| h vi –vj, u i | ¸ 0.01
d
Obs: Deleted pairs are stretched and they form a matching.
Next 10-12 min: Proof-sketch of Structure Thm
( algorithm to produce  -separated S, T of size (n);
 = 1/ log n )
T
S
“Matching is of size o(n) whp” : naive argument fails
“Stretched pair”: vi, vj such that |vi –vj|2 ·  and
| h vi –vj, u i | ¸ 0.01
d
O( 1 ) £ standard deviation

) PrU [ vi, vj get stretched] = exp( - 1

)
= exp( - log n )
E[# of stretched pairs] = O( n2 ) £ exp(- log n )
Generating a contradiction: the walk on stretched pairs
Contradiction if r is
large enough!
Vj
vfinal



Vi
0.01
0.01
d
d
|vfinal - vi| <
r steps
0.01
r
d
| <vfinal – vi, u>| ¸ 0.01r
d
= O( r ) x standard dev.

u
r
Measure concentration (P. Levy, Gromov etc.)
A : measurable set with (A) ¸ 1/4
<d
A
A : points with distance ·  to A
(A) ¸ 1 – exp(-2 d)
A
Reason: Isoperimetric inequality for
spheres

Expander flows
(approximate certificates of expansion)