Geometry and Expansion: A survey of recent results

1
Geometry and Expansion A survey of recent results

• Sanjeev Arora Princeton

( touches upon S. A., Satish Rao, Umesh
Vazirani, STOC04 S. A., Elad Hazan, and
Satyen Kale, FOCS04 S. A., James Lee, and
Assaf Naor, STOC05 papers that are not
mine)
2
Sparsest Cut / Edge Expansion
G (V, E)
c- balanced separator
Both NP-hard
3
The three main characters
Expansion
Isoperimetry (continuous analog of expansion)
Geometry (and geometric embeddings of finite
metric spaces)
4

• 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

5
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 Shmoys95)
• Discrete analog of isoperimetry useful in
  Riemannian geometry (via 2nd eigenvalue of
  Laplacian (Cheeger70)
• Graph-theoretic parameters of inherent interest
  (cf. Lipton-Tarjan planar separator theorem)

6
Previous approximation algorithms

1. Eigenvalue approaches (Cheeger70, Alon85,
   Alon-Milman85)Only yield factor n
   approximation. 2c(G) ? (G) c(G)2 /2

2) O(log n) -approximation via LP (multicommodity
flows)

(Leighton-Rao88)

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

(Linial, London, Rabinovich94,
AR94)
7
New results of ARV04

1. O( ) -approximation to sparsest cut
   and conductance
2. O( )-pseudoapproximation to c-balanced
   separator (algorithm outputs a c-balanced
   separator, c lt c)
3. Existence of expander flows in every graph
   (approximate certificates of expansion)

Disparate approaches from previous slide get
unified
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• 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

Next Semidefinite relaxations for c-balanced
separator (and how to round the solution)
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c-balanced separator
Semidefinite relaxation for
vi vj2/4 1
vi vj2 0
1
S
-1
Find unit vectors
cut semimetric
in ltn
Assign 1, -1 to v1, v2, , vn to
minimize ?(i, j) 2 E vi
vj2/4 Subject to ?i lt j vi vj2/4
c(1-c)n2
Triangle inequality
vi vj2 vj vk2 vi vk2 8 i, j, k
10
Unit l22 space
Unit vectors v1, v2, vn 2 ltd
vi vj2 vj vk2 vi vk2 8 i, j, k
non obtuse !
Example Hypercube -1, 1k
u v2 ?i ui vi2
2 ?i ui vi 2 u v1
In fact, l2 and l1 are subcases of l22
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Structure Theorem for l22 spaces ARV04
Subsets S and T are ?-separated if
for every vi 2 S, vj 2 T vi vj2 ?
ltd
?
Thm If ?ilt j vi vj2 ?(n2) then 9
S, T of size ?(n) that are ?
-separated for ? ?( 1 )
12
Main thm ) O( )-approximation
log n
v1, v2,, vn 2 ltd is optimum SDP soln
SDPopt ?(i, j) 2 E vi vj2
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 vj2 E(R, Rc) ?
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Other new -approximation algorithms

• MIN-2-CNF deletion and several graph deletion
  problems. Agarwal, Charikar, Makarychev,
  Makarychev04
• MIN-LINEAR ARRANGEMENT Charikar, Karloff,
  Rao04
• 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)
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• 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

(Algorithm to produce ? -separatedsets S, T, of
size ?(n) )
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Algorithm to produce two ? separated sets
ltd
Easy Su and Tu likely to have size ?(n)
u
Tu
Delete any vi 2 Su, vj 2 Tu s.t. vi vj2 lt ?.
(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.
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Naïve analysis of random projection fails
v
ltd
u
ltu, vgt ??
standard deviations
E of stretched pairs n2 exp(-?) À n
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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
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Walks in unit l22 space
Unit vectors v1, v2, vn 2 ltd
vi vj2 vj vk2 vi vk2 8 i, j, k
Angles are non obtuse
Taking r steps of length s
only takes you squared distance rs2 (i.e.
distance r s)
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Proof by contradiction (contd.)

Claim 9walk on stretched edges
VERY UNLIKELY IF r large enough) Walk
impossible (CONTRADICTION)
Stretched pair vi vj2 lt ? ltvi vj, ugt
0.01
d
?
?
?
.
u
Why walk is possible delicate argument measure
concentration
vfinal v0 r ?
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• 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.
• Geometric embeddings of metric spaces
• Introduction to expander flows and O(n2) time
  algorithms
• Open problems

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Finite metric space (X, d)
f(x)
y
f

d(x,y)

x
f(y)
distortion of f is minimum Cgt1 such that d( x,
y) f(x ) f( y)2 C d( x, y) 8 x, y
Thm (Bourgain85) For every n-point metric
space, a map
exists with distortion O(log n)
LLR94 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 ?
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Embeddings and Cuts (LLR94, AR94)
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
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Status report of this area
Best upperbound
Best lowerbound
l1 into l2 log0.5 n Enflo69
l22 into l1 1.16 Zatloukal04 Superconstant Khot, Vishnoi04
l22 into l2 log0.5 n Enflo69
Disproves Goemans-Linial conjecture
log n Bourgain85
Uses fourier techniques developed for PCPs!
log0.75 n Chawla,Gupta,Racke 04
Exactly the integrality gap of SDP for general
SPARSEST CUT LLR94, AR94
log0.5 n log log n A., Lee, Naor04
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UpperboundsFrechets 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, Ais are chosen using
S.T.and Measured descent idea of
Krauthgamer, Lee, Naor, and Mendel04
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Embedding lowerbounds (Khot-Vishnoi05)
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)
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• 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

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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)
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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)
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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.)
• Combinatorial approximation algorithms for
  other problems ? (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-Rao88 O(log n) approximation)
• Other applications of expander flows?
  (Useful in some geometric results Naor, Rabani,
  Sinclair04)

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Looking forward to more progress
Thanks !
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New Result (A.,
Hazan, KaleFOCS04)
O(n2) time algorithm that given any graph G finds
for some D gt0

• 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.
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Expander flows LP view
1
LP feasible ) ? ?(D)
D
Thm ARV 9 ?0 s.t. the LP is feasible with D
?/vlog n
G
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Open problems (circa April04)
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
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Various new results
O(n2) time combinatorial algorithm for sparsest
cut (does not use semidefinite programs)
A., Hazan, Kale04
New results about embeddings (i) lp into lq J.
Lee04
(ii) l22 and l1 into l2 CGR04 (approx for
general sparsest cut)
Clearer explanation of expander flows and their
connection to embeddings NRS04
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Formal statement 9 ?0 gt0 s.t. foll. 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
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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/ ?)

• 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 ?
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Example of l22 space hypercube -1, 1k
u v2 ?i ui vi2
2 ?i ui vi 2 u v1
In fact, every l1 space is also l22
Conjecture (Goemans, Linial) Every l22 space is
l1 up to
distortion O(1)
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LP Relaxations for c-balanced separator
Semidefinite
Min ?(i, j) 2 E Xij
0 Xij 1
Motivation Every cut (S, Sc) defines a
(semi) metric
Xij 2 0,1
Xij Xj k Xik
? ilt j Xij c(1-c)n2
There exist unit vectors v1, v2, , vn 2 ltn
such that Xij vi - vj2 /4
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Semidefinite relaxation (contd)
Min ?(i, j) 2 E vi vj2/4 vi2 1 vi
vj2 vj vk2 vi vk2 8 i, j, k ?i lt
j vi vj2 4c(1-c)n2
Unit l22 space
Many other NP-hard problems have similar
relaxations.
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Algorithm to produce two ? separated sets
ltd
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.
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Next 10-12 min Proof-sketch of Structure Thm
( algorithm to produce ? -separated S, T of size
?(n) ? 1/ )
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Matching is of size o(n) whp naive argument
fails
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Generating a contradiction the walk on stretched
pairs
Contradiction if r is large enough!
Vj
vfinal
?
?
?
Vi
r steps
u
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Measure concentration (P. Levy, Gromov etc.)
ltd
A measurable set with ?(A) 1/4
A? points with distance ? to A
A
?(A?) 1 exp(-?2 d)
A?
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Expander flows (approximate certificates of
expansion)