expander flows, graph spectra and graph separators umesh vazirani u.c. berkeley based on joint work...
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Expander Flows, Graph Spectra and Graph Separators
Umesh VaziraniU.C. Berkeley
Based on joint work with Khandekar and Rao and with Orrechia, Schulman and Vishnoi
Graph Separators
S T
Sparsest Cut/Edge Expansion: S
EG
TS
VS
,
2
||||
min)(
c-Balanced Separator: S
EG
TS
VSVc
,
2
||||||
min)(
Applications
• Clustering• Image segmentation• VLSI layout
• Underlie many divide-and-conquer graph algorithms
Interesting Techniques
• Spectral methods. Connection to differential
geometry, discrete isoperimetric inequalities.
• Linear/semidefinite programming
• Measure concentration
• Metric embeddings
Geometrical view
• Map vertices to points in some abstract space: - points well-spread - edges short
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
Spectral Method
1)(
)(min
2
,
2
,
jji
i
jEji
i
xx
xx
Cut at random
Minimize sum of “edge lengths”:
Spread out vertices:
[Cheeger’70] [Alon, Milman ’85][Jerrum, Sinclair’89]
4
)()(
2
10
GG
Leighton-Rao ‘89
mw
w
jiij
Ejiij
,
,
min
22
22
2
5
4
5
5
4
11
1
1
1
1
Cut along ball of random radius
Distances form a metric: satisfy triangle inequality.wij + wjk >= wik
Minimize sum of “edge lengths”:
Spread out vertices:
O(log n) approximation: Approximate max-flow min-cut thm for multi-commodity flows.
ARV ‘04
Triangle inequality:222 )()()( kikjji vvvvvv
Unit sphere in Rd
Unit L22 embedding:
No anglesobtuse
Minimize sum of “edge lengths”
Spread out vertices
22
,
2
,
)1(4)(
)(min
nccvv
vvW
jji
i
jEji
i
)log)(( nGO Procedure to recover cut of size
ARV Procedure to recover cut
• Slice a randomly oriented “fat”-hyperplane of width
d
O1
Unit sphere in Rd
ARV Procedure to recover cut
• 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:
d
O1
Unit sphere in Rd
nvu
log
12
S
r0
Metric EmbeddingsFinite Metric Space (X, d)
x
y
Rk with L2 norm
f(x)
f(y)
Distortion of f is min c: ),()()(),(2
yxdcyfxfyxd
[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 nnO logloglog
f()
nlog
Today’s Talk
• 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*(min{n1.5, n/α(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
Multi-commodity flow:
Single commodity flow:
Expander Flows
• Any algorithm for approximating sparse cuts must find a good cut, of expansion say β
• Must also certify no cut is much smaller.
• To give a k-approximation must certify that no cut has expansion less than β/k.
• Problem: there are exponentially many cuts.
ST
Expander Flows
G = H =
• For each edge of H, route one unit of flow through G
Expander Flows
G = H =
• For each edge of H, route one unit of flow through G
• Must route Ώ(|S|) units of flow from S to T.
• Therefore |ES,T| = Ώ(|S|/c) expansion = Ώ(1/c)
• Ideally c = O(1/α(G))
ST
max congestion = c expansion = Ώ(1/c)
Expander Flows
• max congestion = c. expansion = Ώ(1/c).
• ARV: max congestion =
• Leighton-Rao: H = complete graph. max cong = O(logn/α(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
)(
log
G
nO
• Know large number of vertices on each side of cut.
• A max-flow, min-cut computation should reveal sparse cut.
• But this is circular…
KRV
s t
• H Φ
• 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.
Outline of Algorithm
Cut-Matching Game
H Φ
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 untilH is an expander
Claim: There is a cut player strategy that succeeds in O(log 2 n) rounds.
Finding a cut: Spectral-like-method
= +1 charge
= –1 charge
Mix the charges alongthe matchings { M1, M2, …, Mt }
Random assignment of charge
V: Vertex setx
y
(x+y)/2
(x+y)/2
After t iterations, H = { M1, M2, …, Mt }.
Finding a cut: Spectral-like-method
Order the vertices according to the final charge presentand cut in half.
n/2 n/2
S S
But how to formalize intuition?
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.
Walk Embedding
H Rn,
Vertex i mapped to Pi = (pi1, …, pin)
pij = P[walk started at j ends at i]
Ht = { M1, M2, …, Mt }.
Small cut in graph shows up as clusters in walk embedding. (1/n, …, 1/n)
P1
P3P2
Pn
2
,
2)/1)((/)()( ntpntPt
jiij
ii 1Potential:
Claim: ψ(t) ≤1/4n2 implies α(Ht)≥ ½
Will show potential reduces by (1 – 1/log n) in each iteration.
Ψ(0) = n-1
(1/n, …, 1/n)
P1
P3P2
Pn
Main Question: How to augment Ht = { M1, M2, …, Mt }
by Mt+1 so H closer to expander?2
,
2)/1)((/)()( ntpntPt
jiij
ii 1Potential:
If Mt+1 matches vertex u to vertex v,
then potential reduction in t+1-st step
Since each of Pu and Pv replaced by
So potential reduction =
2vu PP
22
22
2
1
22 vu
vuvu PP
PPPP
The Lifted Walk
2
2
1vu PP
Potential Reduction
Pv
= v |Pv1/n|2
Reduction in = |green|2
1-d: reduction = ()n-d 1-d: log n stretch
Actual potential reduction = /log n
Original random walk = projection of lifted walk on random vector
Running time
• Number of iterations = O(log2 n)• Each iteration = 1 max-flow + O*(n) work
= O*(m3/2)
• [Benczur-Karger’96] In O*(m) time, we can transform any graph G on n vertices into G’ on same vertices:– G’ has O(n log (n)/ε2) edges– All cuts in G’ have size within (1 ± ε) of those in G
• Overall running time = O*(m + n3/2)
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 Mt+1 perform t steps of natural random walk
(instead of round-robin walk) on Ht = { M1, M2, …, Mt }
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.
Why natural walk?
Suppose round robin walk on M1, … , Mk mixes perfectly
on each of S, T. Now a single averaging step on Mk+1
ensures perfect mixing on entire graph!
S T
Mk+1
Matrix inequality: tttt ABAABA )(
Question: Replace ½ self-loop with a ¾ self-loopin round-robin random walk!
x
y
(3x/4+y/4)
(3y/4 + x/4
Gives a way of relating round robin walk to time independent walk.
Conclusions and Open Questions
• Our algorithm is very similar to some heuristics.
• [Lang’04] similar to one iteration of our algorithm.
• METIS [Karypis-Kumar’99]– 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?
When the Expansion is large …
• Could have used [Spielman-Teng’04] “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.
• [Khot, Vishnoi] Ώ(loglog n) integrality gap
• [Orrechia, Schulman, V, Vishnoi] Ώ(√logn)bound on cut-matching game.
• Is it possible to obtain a O(√log n) approximation algorithm using single commodity flows via the cut-matching game?
Limits to these methods
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