What have we learnt about graph expansion
in the new millenium?
Sanjeev Arora
Princeton University &
Center for Computational Intractability
Overview
Last millenium:
•
•
•
•
Central role of expansion and expanders
Recognizing expander graphs via eigenvalues (Cheeger71,Alon-Milman85)
O(log n)-approximation via flows (Leighton-Rao88); region-growing technique;
Close connection to metric embeddings; O(log n) approximation for general
sparsest cut via Bourgain’s Embedding Theorem (Linial-London-Rabinovich, Aumann-Rabani)
This millenium (so far):
• O(√log n )-approximation (A., Rao, Vazirani 04) via both SDP and flows;
• Better metric embeddings; O(√log n )-approximation for general sparsest cut
(Chawla-Gupta-Raecke05, A.-Lee-Naor06)
• Inapproximability results via Unique Games Conjecture (CKKRS06; KV06)
• Lowerbounds for metric embeddings (inspired by PCPs) [KV06; others]; lowerbounds
for SDPs;
• Progress in relating full eigenvalue spectrum to (small-set) expansion (A., Barak,
Steurer10)
(Will not talk about: New understanding of expansion in Cayley graphs of groups, new
algebra-free constructions of optimal expanders, etc.)
Graph Expansion
Important concept: derandomization, network routing, coding theory,
Markov chains, differential geometry, group theory
d-regular graph G
d
vertex set S
expansion(S) =
# edges leaving S
d |S|
a-expander: expansion(S) ≥ a for all S (co-NP-hard to recognize)
α2/2 ≤ λ ≤ 2α [Cheeger, Alon 85, Alon-Milman85].
λ= smallest nonzero eigenvalue of Laplacian of G.
(often us
willtorestrict
attention
sets: |S|, |Sc| > W(n))
Allows
recognize
graphswlog
withtoα “balanced”
= Ω(1) (“expander”)
Approximating expansion via flows
[Leighton-Rao’88]
O(log n)-approximation.
Find largest b s.t. we can
simultaneously route b/n units of
flow between every vertex pair.
(“embed a complete graph”)
b / 2 £ a £ O(logn)b
β⁄n units of flow
bet. each vtx pair
Why α ≥ β/2 :
Total flow out of each subset S
is β⁄n × |S| (n - |S|) ≥ β|S|/2
Why α ≤ O(log n) β:
The LP expressing existence of flow
is feasible if graph diameter is O(1/β).
(uses duality theorem)
In a graph with expansion α,
diameter is O(log n/α).
(Region growing argument:
BFS from S one step at a time;
# of edges increases by (1+α) factor each step;
reach >1/2 the edges in O(log n/α) steps.)
Approximating expansion via expander flows
(A.,Rao, Vazirani 2004)
β units of flow
originating at
each vtx
Route a flow with some demand graph W= (wij)
(wij = flow between i and j)
s.t. W is β-regular and has expansion 0.01
(“expander flow”)
Maximise β.
Easy: α ≥ 0.01 β
(Amount of flow leaving each set S is at least 0.01 β |S|.)
Main claim: α≤ O(β √log n)
Next: Geometry of cuts and how efficiently they can be crossed
Geometry of cuts
Cut semimetric
dS(i,j) = 1 if i, j on opposite sides of the cut,
= 0 else.
1
(gives embedding into a line)
0
Convex combination of cut semimetrics
d(i, j) = ΣS αS dS(i, j)
(Gives embedding into l1 : i  vi
|vi – vj|1= fraction of cuts i, j are across)
Approximating expansion via flows (A.,Rao, Vazirani 2004)
β units of flow
originating at
each vtx
LP
Route
formulation:
a flow with demand graph W= (wij)
(wij = flow between i and j)
W is β-regular and has expansion 0.01
Maximise β.
Check by computing
eigenvalue (“separation
oracle”)
Main claim: α≤ O(β √log n)
Duality Thm 
Feasibility follows if for every distribution (αS) on balanced cuts,
there are Ω(n) disjoint vertex pairs (i1, j1), (i2, j2), … s.t.
(a) d(ir, jr) = O(√log n/ α)
1st structure theorem
(b) ir, jr are across Ω(1) fraction of cuts.
Open: Replace √log n with o(√log n )? (Best lowerbound: log log n [DKSV06])
[ARV04] If G is an α-expander then for every distribution (αS)
on balanced cuts, there are Ω(n) disjoint vertex pairs (i1, j1),
(i2, j2), … s.t.
(a) d(ir, jr) = O(√log n/ α)
(b) ir, jr are across Ω(1) fraction of cuts.
Thoughts on
Structure Thm
sink
1
1
Warmup: If max degree= O(1) and given a single
balanced cut, above is true with O(1/α) instead of
O(√log n/ α)
Pf: Max-Flow Min Cut Thm
4/α
1
1
source
(all other edges
capacity 4/α )
α-expansion  Min Cut = Ω(|S|) = Ω(n) =Max-Flow
Total capacity = O(n/α)
Flow decomposition  Ω(n) flowpaths of length O(1/α)
with one endpoint in S and one in Sc
A flow-based O(√log n)-approximation algorithm for expansion
For β = 1/n, 2/n, 4/n,… do
Try to solve above LP to route a β-regular expander flow in G
If succeed, have verified that expansion ≥ 0.01β
If fail, use [ARV] technique to find a cut of expansion < O(β√log n)
(note: before finding this cut had already verified expansion ≥ 0.01 β/2)
(Note: Can be implemented in O(n1.5) time using matrix multiplicative
weight method [Sh09,AK07,AHK05]. Satyen Kale’s talk.)
Suggested research directions
Nothing special about routing an expander flow; could use any graph
family whose expansion can be verified up to O(1) factor. (Suffices to
solve LP.)
Example: Graphs with a few small nonzero eigenvalues (generalizes expanders,
which have no small eigenvalues) [A., Barak, Steurer’10]
Could also try for o(√log n) approximation in subexponential time.
See David Steurer’s talk….
View 2: Use of math programming relaxations
Semidefinite
Linear
Integer program for c-balanced separator (expansion of sets of size ≥ cn)
Recall:
[LR88]; O(log n)-approximation
Cut semimetric
dS(i,j) = 1 if i, j on opposite sides of the cut,
= 0 else.
[ARV04]: O(√log n) –approximation.
(Main obstacle: understanding vectors satisfying triangle inequality condition).
How to round the SDP: 2nd Structure Theorem
v1, v2, v3, … : unit vectors in Rn, s.t.
avg |vi –vj|2 = Ω(1) (“well-spread”)
|vi –vj|2 + |vj-vk|2 ≥ |vi –vk|2
(l22 property; angle subtended by any two points
on the third is nonobtuse; includes l1 as subcase)
THM: For Δ = Ω(1/√log n) there exist sets S, T of size Ω(n) s.t.
|vi –vj|2 ≥ Δ for all i in S, j in T
Δ
(S, T are Δ-separated)
NB: Implies weak version of 1st Structure Thm: Maxflow Mincut applied to S, T
yields Ω(n) paths of length O(1/α) that cross Ω(1/√log n) fraction of cuts.
Rounding the SDP
S, T: Δ-separated sets of size Ω(n)
Do BFS wrt distance function d(i,j) = |vi –vj|2 ,
starting from S and going until you hit T
Output the level of the BFS tree with least expansion.
Claim: This gives a balanced cut (O, Oc)
s.t. |E(O, Oc)| ≤ SDPOPT /Δ = O(√log n) SDPOPT
d(S, i)
d(S, j)
S
vj
vi
d(S, j) – d(S, i) ≤ |vi –vj|2
Edge (i,j) contributes to cut for ≤ |vi –vj|2 levels,
and each level cuts at least |E(O, Oc)| edges.
O(√log n)-approximation for other cut-like problems
•
MIN-2-CNF deletion and several graph deletion problems. [Agarwal, Charikar,
Makarychev, Makarychev04]
[’04]. Weighted version of S
•
MIN-LINEAR ARRANGEMENT [Charikar, Karloff, Rao’05]
•
General SPARSEST CUT [Chawla-Gupta-Raecke05, A. Lee Naor’06] 0
•
Min-ratio VERTEX SEPARATORS and Balanced VERTEX SEPARATORS[Feige, Hajiaghayi, Lee’04]
•
•
All
Method: SDP rounding using a generalized structure theorem…
Suggested future direction
SDP with triangle inequality corresponds to level 2 of
Lasserre, Lovasz-Schrijver, etc. (see Madhur Tulsiani’s talk)
Use more powerful SDP relaxations from higher levels?
* May need to allow superpolynomial time (rth level  nr time)
* Not currently ruled out under UGC.
Cut problems and embeddings
General Sparsest Cut: Cost matrix (cij) cij ≥0;
Demand matrix (dij) dij ≥ 0;
Find
SDP relaxation:
[LLR94,AR94]: Integrality gap
= Minimum distortion incurred
when embedding l22 metrics into
l1 (= convex combination
of cut semi-metrics)
Geometric Embeddings of Metric Spaces
Finite metric space (X, d)
Geometric space, eg l1
f
f(x)
x
d(x,y)
y
f(y)
Distortion of f : Minimum C s.t. d(x, y) ≤ |f(x) –f(y)| ≤ C d(x,y)
[Bourgain’85, LLR94]: Distortion O(log n) into l1, l2
What if X is itself geometric?
[Chawla-Gupta-Raecke05, A.-Lee-Naor06]: Distortion O(√log n  log log n) for
embedding l22 into l1; and embedding l1 into l2
Embedding theorems in one slide
Tool 1: Padded decompositions [Krauthgamer,Lee, Mendel,Naor04]
Scale S, padding parameter p:
Metric
space
(X, d)
Line embedding
Partition probabilistically into pieces
of diameter ≤ S, s.t. for all x
Pr[x’s partition contains Ball(x, S/p)] ≥ ½
x

0
Map each block to 0 with probability ¼;
Map x to d(x, zero-block)
Tool 2: Use ARV structure theorem to produce padded decompositions
at different scales; combine line embeddings into a single embedding using
“measured descent.”
Proving lowerbounds on distortion
• [Khot-Vishnoi05] log log n lowerbound; construction inspired by PCPs
(hypercontractivity of noisy hypercubes)
• [Lee-Naor],[Cheeger,Kleiner,Naor] (log n)ε lowerbound; construction
based upon Heisenberg group; new notion of differentiation
• [Lee-Muharrami] √log n lowerbound; only for embedding
weak l22 spaces into l1. Elementary construction and analysis.
Open: √log n lowerbound for l22 spaces;
(log n)ε lowerbound for SDP integrality gap of uniform sparsest cut
(ie edge expansion).
Proof of Structure Theorems
Recall:
v1, v2, v3, … : unit vectors in Rd, s.t.
avg |vi –vj|2 = Ω(1) (“well-spread”)
|vi –vj|2 + |vj-vk|2 ≥ |vi –vk|2
For Δ = Ω(1/√log n) there exist sets S, T of size
Ω(n) s.t.
|vi –vj|2 ≥ Δ for all i in S, j in T
(S, T are Δ-separated)
Δ
(Recall: 1st Structure Theorem concerned distributions of cuts,
which correspond to l1 metrics, which are a subcase of l22)
Algorithm to produce two Δ-separated sets (Δ= 1/√log n)
0.01/√d
Easy: Su and Tu are likely to have size Ω(n).
Tu
Delete any vi in Su and vj in Tu s.t. |vi – vj|2 < Δ
(repeat until no such pair remains)
Su
u
If Su and Tu still have size Ω(n) output them.
Main difficulty: Show that whp only o(n) points get deleted.
Obs: Deleted pairs were “stretched”, i.e.,
|vi – vj|2 < Δ,
|<vi – vj, u>| > 0.01/√d
Fact: Pr[|<vi – vj, u>| > 0.01 √Δ/√d]
= exp(-1/Δ) = exp(-√log n). Too large for union bound
Walks in l22 space
|vi –vj|2 + |vj-vk|2 ≥ |vi –vk|2
Δ
Δ
Δ
Δ
r steps of squared-length Δ only take you
a total squared distance rΔ (i.e., distance √r Δ)
Main proof step: Use measure concentration to prove that for most directions
u there is a walk of length r on stretched edges (v1, v2), (v2, v3),.. (vr, vr+1)
so that |<v1 – vr+1, u>| > 0.001 r/√d
Pr[such v1 , vr+1 exist in the point set] < exp(- r/Δ) < 1/n2
Unique Games Conjecture
[Khot03] Given m equations in n variables x1, x2, …, xn of the type
axi + b Xj = a (mod 113)
s.t. (1-ε) fraction are simultaneously satisfiable, it is NP-hard to satisfy
½ of them simultaneously.
Used to prove best inapproximability results for host of problems, including
expansion problems. Inspired SDP integrality gaps (aka embedding lowerbounds).
(See Khot’s talk)
(Expansion strikes back) The Achilles heel of UGC appears to be expansion.
Better understanding of small-set expansion may disprove UGC.
(see Steurer’s talk)
Looking forward to more insight in the next decade!
Thank you!