separator

1
Expander Flows, Graph Spectra and Graph
Separators
Umesh Vazirani U.C. Berkeley
Based on joint work with Khandekar and Rao
and with Orrechia, Schulman and Vishnoi
2
Graph Separators
S
T
Sparsest Cut/Edge Expansion
c-Balanced Separator
3
Applications

• Clustering
• Image segmentation
• VLSI layout
• Underlie many divide-and-conquer graph
  algorithms

4
Interesting Techniques

• Spectral methods. Connection to differential
• geometry, discrete isoperimetric inequalities.
• Linear/semidefinite programming
• Measure concentration
• Metric embeddings

5
Geometrical view

• Map vertices to points in some abstract space
• - points well-spread
• - edges short

6
Geometrical view

• Map vertices to points in some abstract space
• - points well-spread
• - edges short
• Good bisection of the space yields sparse cut
  in graph

7
Spectral Method
Cut at random
Minimize sum of edge lengths Spread out
vertices
Cheeger70 Alon, Milman 85Jerrum,
Sinclair89
8
Leighton-Rao 89
1
4
1
1
5
2
1
2
1
2
5
2
2
4
1
5
Cut along ball of random radius
Minimize sum of edge lengths Spread out
vertices
Distances form a metric satisfy triangle
inequality. wij wjk gt wik
O(log n) approximation Approximate max-flow
min-cut thm for multi-commodity flows.
9
ARV 04
Unit sphere in Rd
No angles obtuse
Unit L22 embedding
Triangle inequality
Minimize sum of edge lengths Spread out
vertices
Procedure to recover cut of size
10
ARV Procedure to recover cut
Unit sphere in Rd

• Slice a randomly oriented fat-hyperplane of
  width

11
ARV Procedure to recover cut
Unit sphere in Rd
S

• Slice a randomly oriented fat-hyperplane of
  width
• Discard pairs of points (u,v)
• Arrange points according to distance from S
• Cut along ball of random radius r

12
Metric Embeddings
Finite Metric Space (X, d)
Rk with L2 norm
y
f(y)
f()
x
f(x)
Distortion of f is min c
Bourgain 85 Every finite metric space can be
embedded in L2 with distortion O(log n).
Longstanding open question Better bound for
L1? Enflo 69 Arora, Lee, Naor 05 Any
finite L1 metric can be embedded in
L2 with distortion
13
Todays Talk
Multi-commodity flow

• Leighton-Rao multi-commodity flow O(n2).
• Arora, Hazan, Kale O(n2) ARV implementation
• based on expander-flow formalism
• Much faster in practice.
• Khandekar-Rao-V O(minn1.5, n/a(G))
  single commodity
• flow based algorithm. O(log2 n) approx. ratio.
• Arora, Kale matrix multiplicative weights
  algorithm
• based O(log n) approx
• Orrechia, Schulman, V, Vishnoi O(log n)
  approx using
• KRV style algorithm

Single commodity flow
14
Expander Flows

• Any algorithm for approximating sparse cuts
  must
• find a good cut, of expansion say ß
• Must also certify no cut is much smaller.
• To give a k-approximation
• must certify that no cut
• has expansion less than
• ß/k.
• Problem there are exponentially many cuts.

T
S
15
Expander Flows
G
H

• For each edge of H, route one unit of flow
  through G

16
Expander Flows
T
G
H
S

• For each edge of H, route one unit of flow
  through G
• Must route ?(S) units of flow from S to T.
• Therefore ES,T ?(S/c) expansion
  ?(1/c)
• Ideally c O(1/a(G))

max congestion c expansion ?(1/c)
17
Expander Flows

• max congestion c. expansion ?(1/c).
• ARV max congestion
• Leighton-Rao H complete graph. max cong
  O(logn/a(G))
• tight example G expander graph.
• Motivating idea for ARV write LP to find best
  embedding
• of H in G exponentially many constraints
  saying H expander
• eigenvalue bound gives efficient test for
  expansion!
• Therefore poly time using Ellipsoid algorithm.
• Arora, Hazan, Kalle O(n2) implementation of
  ARV

18
KRV
t
s

• Know large number of vertices on each side of
  cut.
• A max-flow, min-cut computation should reveal
  sparse cut.
• But this is circular

19
Outline of Algorithm

• H F
• Embed candidate expander H in G with small
  congestion.
• Test whether H is expander (if so done!)
• Else non-expanding cut in H gives a bipartition
  of G
• route a flow in G across this bipartition.
• Decompose flow into flow paths and add the
  resulting
• matching to H.

20
Cut-Matching Game
H F

• Cut Player
• Find bad 50-50 cut in H
• Goal min iterations until
• H is an expander

• Matching Player
• Pick a perfect matching
• across cut
• Goal max iterations until
• H is an expander

Claim There is a cut player strategy that
succeeds in O(log 2 n) rounds.
21
Finding a cut Spectral-like-method
After t iterations, H M1, M2, , Mt .
1 charge
Random assignment of charge
1 charge
(xy)/2
x
V Vertex set
Mix the charges along the matchings M1, M2, ,
Mt
y
(xy)/2
22
Finding a cut Spectral-like-method
Order the vertices according to the final charge
present and cut in half.
S
S
n/2
n/2
But how to formalize intuition?
23
Lift to Rn

• Cannot directly formalize previous intuition
• Therefore lift random walk to Rn walk
  embedding of H.
• n-dimensional vector associated with each
  vertex
• In each step, replace vectors at endpoints of
  matched
• edge by their average vector.
• Potential function to measure progress of this
  process.
• Potential function small implies H expander.
• Relate lifted process to original random walk
• each successive matching decreases potential
  function.

24
Walk Embedding
Ht M1, M2, , Mt .
H Rn,
Vertex i mapped to Pi (pi1, , pin) pij
Pwalk started at j ends at i
P3
P2
P1
Small cut in graph shows up as clusters in
walk embedding.
(1/n, , 1/n)
Pn
Potential
?(0) n-1
Claim ?(t) 1/4n2 implies a(Ht) ½ Will show
potential reduces by (1 1/log n) in each
iteration.
25
The Lifted Walk
Main Question How to augment Ht M1, M2, ,
Mt by Mt1 so H closer to expander?
P3
Potential
P2
If Mt1 matches vertex u to vertex v, then
potential reduction in t1-st step Since each
of Pu and Pv replaced by So potential reduction

P1
(1/n, , 1/n)
Pn
26
Potential Reduction
Original random walk projection of lifted walk
on random vector
Pv
? ?v Pv?1/n2 Reduction in ? ?green2
1-d reduction ?(?) n-d ? 1-d log n stretch
?
Actual potential reduction ?/log n
27
Running time

• Number of iterations O(log2 n)
• Each iteration 1 max-flow O(n) work
• O(m3/2)
• Benczur-Karger96 In O(m) time, we can
  transform any graph G on n vertices into G on
  same vertices
• G has O(n log (n)/e2) edges
• All cuts in G have size within (1 e) of those
  in G
• Overall running time O(m n3/2)

28
Improving to O(log n) approximation

• Arora, Kale matrix multiplicative weights
  algorithm based
• combinatorial primal-dual schema for
  semidefinite progs
• Orrechia, Schulman, V, Vishnoi simple KRV
  style algorithm

Idea To find Mt1 perform t steps of natural
random walk (instead of round-robin
walk) on Ht M1, M2, , Mt
29
Brief Sketch

• Instead of showing that H has constant edge
• expansion after O(log2 n) steps, will show
  that
• the spectral gap of H is at least 1/log n, and
  therefore
• the conductance of H is at least 1/log n.
• Since degree of H is log2 n, this means its
  edge
• edge expansion is at least log n.

30
Why natural walk?
Mk1
S
T
Suppose round robin walk on M1, , Mk mixes
perfectly on each of S, T. Now a single averaging
step on Mk1 ensures perfect mixing on entire
graph!
31
Matrix inequality
Gives a way of relating round robin walk to time
independent walk.
Question Replace ½ self-loop with a ¾
self-loop in round-robin random walk!
(3x/4y/4)
x
y
(3y/4 x/4
32
Conclusions and Open Questions

• Our algorithm is very similar to some heuristics.
• Lang04 similar to one iteration of our
  algorithm.
• METIS Karypis-Kumar99
• collapses random edges
• finds a good partition in collapsed graph
• induces it up to original graph, using local
  search
• Connections with these heuristics? Rigorous
  analysis?

33
When the Expansion is large

• Could have used Spielman-Teng04 nibble
  algorithm instead of walk-embedding. But

Algorithm Output sparsity Running Time
Spectral ?1/2 n2/?2
Spielman-Teng ?1/3 log3 n n/?3
KRV ? log2 n min n3/2,n/?

• Conjecture A single iteration of round-robin
  walk
• max-flow should give a
  sparse cut.

34
Limits to these methods

• Khot, Vishnoi ?(loglog n) integrality gap
• Orrechia, Schulman, V, Vishnoi ?(vlogn)
• bound on cut-matching game.
• Is it possible to obtain a O(vlog n)
  approximation algorithm using single commodity
  flows via the cut-matching game?