Geometry and Expansion: A survey of recent results - PowerPoint PPT Presentation

About This Presentation
Title:

Geometry and Expansion: A survey of recent results

Description:

Previous approximation algorithms for expansion problems ... O( ) -approximation to sparsest cut ... New approximation algorithm via semidefinite programming ... – PowerPoint PPT presentation

Number of Views:51
Avg rating:3.0/5.0
Slides: 46
Provided by: sanj188
Category:

less

Transcript and Presenter's Notes

Title: 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
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)

4
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.
5
Previous approximation algorithms for expansion
problems
  • 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)
6
New results of ARV04
  • O( ) -approximation to sparsest cut
    and conductance
  • O( )-pseudoapproximation to c-balanced
    separator (algorithm outputs a c-balanced
    separator, c lt c)
  • Existence of expander flows in every graph
    (approximate certificates of expansion)

Disparate approaches from previous slide get
unified
7
  • 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)
8
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
9
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
10
Structure Theorem for l22 spaces ARV04
Subsets S and T are ?-separated if
for every vi 2 S, vj 2 T vi vj2 ?
ltd
G? Graph in which (i,j) is an edge iff
vi vj2 ?
?
Thm If ?ilt j vi vj2 ?(n2) then 9
S, T of size ?(n) that are ?
-separated for ? ?( 1 )
Equiv G? is an expander ) ?
11
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) ?
12
Other new -approximation algorithms
Example Structure Theorem (Agarwal, Charikar,
Makarychev2 05)
  • MIN-2-CNF deletion and several graph deletion
    problems. Agarwal, Charikar, Makarychev,
    Makarychev05
  • MIN-LINEAR ARRANGEMENT Charikar, Karloff,
    Rao05
  • General SPARSEST CUT A., Lee, Naor 05
  • Min-ratio VERTEX SEPARATORS and Balanced VERTEX
    SEPARATORS Feige, Hajiaghayi, Lee, 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.)
13
  • 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

14
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 ?
15
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
16
Status report of this area
Best upperbound
Best lowerbound
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
Note l2 µ l1 µ l22
17
Embedding 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)
Why S.T. useful If S obtained from S.T., then
in the mapping x ! d(x, S), many xs (namely,
all those in T) map far from 0.
In recent embeddings, Ais are chosen using
S.T.and Measured descent idea of
Krauthgamer, Lee, Naor, and Mendel04
18
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)
Khot-Naor Lowerbounds for embedding
earth-mover edit metrics into l1
19
  • 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

20
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)
21
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)
22
  • 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

Outline of proof of S. T.
(Algorithm to produce ? -separatedsets S, T, of
size ?(n) )
23
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.
24
Naïve analysis of random projection fails
v
ltd
u
ltu, vgt ??
standard deviations
E of stretched pairs n2 exp(-?) À n
25
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
26
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)
27
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
How to produce walk delicate argument measure
concentration
vfinal v0 r ?
28
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-Rao88 O(log n) approximation)
  • Other applications of expander flows?
    (Useful in some geometric results Naor, Rabani,
    Sinclair04)

29
Looking forward to more progress
Thanks !
30
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.
31
Expander flows LP view
1
LP feasible ) ? ?(D)
D
Thm ARV 9 ?0 s.t. the LP is feasible with D
?/vlog n
G
32
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
33
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
34
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
35
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 ?
36
(No Transcript)
37
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)
38
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
39
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.
40
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.
41
Next 10-12 min Proof-sketch of Structure Thm
( algorithm to produce ? -separated S, T of size
?(n) ? 1/ )
42
Matching is of size o(n) whp naive argument
fails
43
Generating a contradiction the walk on stretched
pairs
Contradiction if r is large enough!
Vj
vfinal
?
?
?
Vi
r steps
u
44
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?
45
Expander flows (approximate certificates of
expansion)
Write a Comment
User Comments (0)
About PowerShow.com