The walk on stretched pairs

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The walk on stretched pairs
Vj
vfinal
?
?
?
Vi
r steps
u
Contradiction!!
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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 Motivation
Expander
G (V, E)
Idea Embed a d-regular (weighted) graph such
that 8 S w(S, Sc) ?(d S)
()
(certifies expansion ?(d) )
S
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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Formal statement 9 ?0 gt0 such that following 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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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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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?

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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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