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Sparsest Cut
SS G) = min |E(S, S)|
|S|S µ V
G = (V, E)
c- balanced separator
G) = min |E(S, S)| |S|
S µ Vc
|S| ¸ c ¢ |V|Both NP-hard
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 Shmoys’95)
• Related to curvature of Riemannian manifolds and 2nd eigenvalue of Laplacian (Cheeger’70)
• Graph-theoretic parameters of inherent interest (cf. Lipton-Tarjan planar separator theorem)
Previous approximation algorithms
1) Eigenvalue approaches (Cheeger’70, Alon’85, Alon-Milman’85)
2c(G) ¸ L(G) ¸ c(G)2/2 c(G) = minS µ V E(S, Sc)/ E(S)
2) O(log n) -approximation via multicommodity flows (Leighton-Rao 1988)
• Approximate max-flow mincut theorems• Region-growing argument
3) Embeddings of finite metric spaces into l1 (Linial, London, Rabinovich’94)
• Geometric approach; more general result
Our results
1. O( ) -approximation to sparsest cut and conductance
2. O( )-pseudoapproximation to c-balanced separator
(algorithm outputs a c’-balanced separator, c’ < c)
3. Existence of expander flows in every graph
(approximate certificates of expansion)
log n
log n
LP Relaxations for c-balanced separator
Motivation: Every cut (S, Sc) defines a (semi) metric
1
1
1
0 0
Xij 2 {0,1}
i< j Xij ¸ c(1-c)n2
Xij + Xj k ¸ Xik
0 · Xij · 1
Semidefinite
There exist unit vectors v1, v2, …, vn 2 <n such that Xij = |vi - vj|2 /4
Min (i, j) 2 E Xij
Semidefinite relaxation (contd)
Min (i, j) 2 E |vi –vj|2/4
|vi|2 = 1
|vi –vj|2 + |vj –vk|2 ¸ |vi –vk|2 8 i, j, k
i < j |vi –vj|2 ¸ 4c(1-c)n2
Unit l22 space
l22 space
Unit vectors v1, v2,… vn 2 <d
|vi –vj|2 + |vj –vk|2 ¸ |vi –vk|2 8 i, j, k
Vi
Vk
Vj
Angles are non obtuse
Taking r steps of length s
only takes you squared distance rs2
(i.e. distance r s)
s ss s
Example of l22 space: hypercube {-1, 1}k
|u – v|2 = i |ui – vi|2 = 2 i |ui – vi| = 2 |u – v|1
In fact, every l1 space is also l22
Conjecture (Goemans, Linial): Every l22 space is l1 up to distortion O(1)
Our Main Theorem
Two subsets S and T are -separated if
for every vi 2 S, vj 2 T |vi –vj|2 ¸
¸
Thm: If i< j |vi –vj|2 = (n2) then there exist two sets S, T of size (n) that are -separated for = ( 1 )
<d
log n
Main thm ) O( )-approximationlog n
v1, v2,…, vn 2 <d is optimum SDP soln; SDPopt = (I, j) 2 E |vi –vj|2
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 –vj|2 ¸ |E(R, Rc)| £
) |E(R, Rc)| · SDPopt /
· O( SDPopt) log n
Next 10-12 min: Proof-sketch of Main Thm
Projection onto a random line
<dv
u
<u, v> ??
1
d
1
d
e-t
2/2
d
log nPru[ projection exceeds 2 ] < 1/n2
Algorithm to produce two –separated sets
<d
u
Su
Tu
0.01
d
Check if Su and Tu have size (n)
If any vi 2 Su and vj 2 Tu satisfy
|vi –vj|2 ·
and repeat until no such vi, vj can be found
delete them
If Su, Tu still have size (n), output them
Main difficulty: Show that whp only o(n) points get deleted
d
“Stretched pair”: vi, vj such that |vi –vj|2 · and | h vi –vj, u i | ¸ 0.01
Obs: Deleted pairs are stretched and they form a matching.
“Matching is of size o(n) whp” : trivial argument fails
d
“Stretched pair”: vi, vj such that |vi –vj|2 · and | h vi –vj, u i | ¸ 0.01
O( 1 ) £ standard deviation
) PrU [ vi, vj get stretched] = exp( - 1 )
= exp( - )log n
E[# of stretched pairs] = O( n2 ) £ exp(- ) log n
Suppose with probability (1) there is a matching of (n) stretched pairs
Vi
Ball (vi , )u
Vj
0.01
d
The walk on stretched pairs
u
Vi
Vj
0.01
d
0.01
d
r steps
0.01
d
r
|vfinal - vi| < r
| <vfinal – vi, u>| ¸ 0.01r
d
= O( r ) x standard dev.
vfinal
Contradiction!!
Measure concentration (P. Levy, Gromov etc.)
<d
A
A : measurable set with (A) ¸ 1/4
A : points with distance · to A
AA) ¸ 1 – exp(-2 d)
Reason: Isoperimetric inequality for spheres
Expander flows: Motivation
G = (V, E)
S S
Idea: Embed a d-regular (weighted) graph such that
8 S w(S, Sc) = (d |S|)
Cf. Jerrum-Sinclair, Leighton-Rao(embed a complete graph)
“Expander”
Graph w satisfies (*) iff L(w) = (1) [Cheeger]
(*)
Our Thm: If G has expansion , then a d-regular expander flow can be routed in it where d=
log n
(certifies expansion = (d) )
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)
Formal statement : 9 0 >0 such that following LP is feasible for d = (G)
log n
fp ¸ 0 8 paths p in G
8i j p 2 Pij fp = d (degree)
Pij = paths whose endpoints are i, j
8S µ V i 2 S j 2 Sc p 2 Pij fp ¸ 0 d |S| (demand graph is
an expander)
8e 2 E p 3 e fp · 1 (capacity)
New result (A., Hazan, Kale; 2004)
O(n2) time algorithm that given any graph G finds for some d >0
• a d-regular expander flow • a cut of expansion O( d )log n
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.
)d) · (G) · O(d )log n
Open problems
• 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
• Any applications of expander flows?
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 =
log n
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