Expander flows, geometric embeddings, and graph partitioning

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Expander flows, geometric embeddings, and graph
partitioning

• Sanjeev Arora Princeton
• Satish Rao UC Berkeley
• Umesh Vazirani UC Berkeley

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Sparsest Cut
G (V, E)
S
c- balanced separator
Both NP-hard
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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 Shmoys95)
• Discrete analogs of isoperimetric constant
  related to curvature of Riemannian manifolds and
  2nd eigenvalue of Laplacian (Cheeger70)
• Graph-theoretic parameters of inherent interest
  (cf. Lipton-Tarjan planar separator theorem)

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Previous approximation algorithms

• Eigenvalue approaches (Cheeger70, Alon85,
  Alon-Milman85)2c(G) ?L(G) c(G)2/2
  c(G) minS µ V E(S, Sc)/ E(S)

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

(Leighton-Rao88)

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

(Linial, London, Rabinovich94,
AR94)
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Our results

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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Semidefinite relaxations for c-balanced
separator (and how to round the solutions)
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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
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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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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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Our Main Theorem
Two 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 there
exist two sets S, T of size ?(n) that are
? -separated for ? ?( 1 )
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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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Next 10-12 min Proof-sketch of Main Thm
( algorithm to produce ? -separated S, T of size
?(n) ? 1/ )
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Projection onto a random line
v
ltd
u
ltu, vgt ??
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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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Matching is of size o(n) whp naive argument
fails
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Suppose matching of ?(n) size exists with
probability ?(1)
.stretched pairs are almost everywhere you look!
Vj
u
Ball (vi , ?)
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Generating a contradiction the walk on stretched
pairs
Vj
vfinal
?
?
?
Vi
r steps
u
Contradiction (if r large enuff)!!
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Measure concentration (P. Levy, Gromov etc.)
ltd
A measurable set with ?(A) 1/4
A
A? points with distance ? to A
?(A?) 1 exp(-?2 d)
A?
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Expander flows (approximate certificates of
expansion)
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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)
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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 (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
Clearer explanation 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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New result (A.,
Hazan, Kale 2004)
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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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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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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