Transcript Approximations for Isoperimetric and Spectral Profile and Related Parameters Prasad Raghavendra MSR New England S joint work with David Steurer Prasad Tetali Princeton University Georgia Tech.

Approximations for Isoperimetric
and
Spectral Profile and Related Parameters
Prasad Raghavendra
MSR New England
S
joint work with
David Steurer
Prasad Tetali
Princeton University
Georgia Tech
Graph Expansion
d-regular graph G
with n vertices
expansion(S) =
# edges leaving S
d |S|
d
vertex set S
A random
neighbor of a random
vertex
Conductance
of Graph
G in S
is outside of S with probability expansion(S)
ФG =
minimum
expansion(S)
|S| ≤ n/2
Extremely well-studied, many different contexts
pseudo-randomness, group theory, online routing, Markov chains, metric
embeddings, …
Uniform Sparsest Cut Problem
Given a graph G compute ФG
and find the set S achieving it.
S
Measuring Graph Expansion
d-regular graph G
expansion(S) =
with n vertices
# edges leaving S
d |S|
d
Conductance of Graph G
vertex set S
ФG =
minimum
expansion(S)
|S| ≤ n/2
Complete
Graph
Path
Complete
Graph
Complete graphs with a perfect matching
Typically, small sets expand to a greater extent.
Isoperimetric Profile
expansion(S) =
S
# edges leaving S
d |S|
Isoperimetric Profile of Graph G
Ф(δ) =
minimum
|S| ≤ δn
expansion(S)
1
• Conductance function – defined by
[Lovasz-Kannan] used to obtain better
mixing time bounds for Markov chains.
Ф(δ)
Set Size δ
0.5
•Decreasing function of δ
Approximating
Isoperimetric Profile
1
expansion(S) =
# edges leaving S
d |S|
Isoperimetric Profile of Graph G
Ф(δ) =
Ф(δ)
minimum
|S| ≤ δn
expansion(S)
S
Set Size
δ
0.5
What is the value of Ф(δ) for a given graph G and a constant
Uniform Sparsest Cut: Determine the lowest point on the curve.
δ > 0?
Gap Small Set Expansion Problem (GapSSE)(η, δ)
Given a graph G and constants δ, η > 0,
Is Ф(δ) < η OR Ф(δ) > 1- η?
----Closely tied to Unique Games Conjecture
(Last talk of the day
“Graph Expansion and the Unique Games Conjecture” [R-Steurer 10])
Algorithm
Theorem
For every constant δ > 0, there exists a poly-time algorithm
that finds a set S of size O(δ) such that
expansion (S) ≤ O( ( ) log(1/  ) )
-- A (Ф(δ) vs
( ) log(1/  ) ) approximation
• For small enough δ, the algorithm cannot distinguish between
Ф(δ) < η OR Ф(δ) > 1- η
Theorem
[R-Steurer-Tulsiani 10]
An improvement over above algorithm by better than constant
factor, would help distinguish Ф(δ) < η OR Ф(δ) > 1- η
A Spectral Relaxation
0
Let x = (x1 , x2 , .. xn) be the indicator
function of the
unknown least expanding set S
SC
2
(
x

x
)
 i j
Number of edges leaving S =
1S
( i , j )E
|S| =  xi2
Relaxing 0,1 to real numbers
 ( )  minn
 (x
i
( i , j )E
 xj )
xx
{0
,1}n
d  xi
|Support(x)| < δn
i
2
i
2
T
x Lx
 min
T
x
{0 ,
1}n d  x x
x
n
Why is it spectral?
Spectral Profile:
[Goel-Montenegro-Tetali]
( )  min
n
2
(
x

x
)
 i j
( i , j )E
x
|Support(x)| < δn
d  xi
2
i
T
x Lx
 min
x n
d  xT x
|Support(x)| < δn
Smallest Eigen Value of Laplacian:
  min
n
x

x 1
2
(
x

x
)
 i j
( i , j )E
d  xi
i
2
xT Lx
 min
T
n
x
dx x

x 1
Observation: Λ(δ) is the smallest possible eigenvalue of a
submatrix L(S,S) of size at most < δn of the Laplacian L .
Rounding Eigenvectors
Smallest Eigen Value of Laplacian
Rounding
x
x T Lx
  m in
n
x
d  xT x

x 1
( )  min
n
x
|Support(x)| < δn
2
(
x

x
)
 i j
( i , j )E
d  xi
i
2
Cheeger’s inequality
There is a sparse cut
of value at most 2
xT Lx
 min
T
x n
dx x
|Support(x)| < δn
0
Lemma:
There exists a set S of volume at most δ whose expansion is at
most 2( )
Spectral Profile
[Goel-Montenegro-Tetali]
( )  min
n
x
|Support(x)| < δn
2
(
x

x
)
 i j
( i , j )E
d  xi
i
2
T
x Lx
 min
x n
d  xT x
|Support(x)| < δn
Unlike eigen values, Λ(δ) is not the optimum of a convex
program. We show an efficiently computable SDP that gives
good guarantee.
Theorem: There exists an efficiently computable SDP
relaxation Λ* (δ) of Λ(δ) such that for every graph G
Λ* (δ) ≤ Λ(δ) ≤ Λ* (δ/2)∙O(log 1/δ)
Recap
Theorem: (Approximating Spectral Profile)
There exists an efficiently computable SDP relaxation Λ* (δ) of
Λ(δ) such that for every graph G
Λ* (δ) ≤ Λ(δ) ≤ Λ* (δ/2)∙O(log 1/δ)
Lemma: (Cheeger Style Rounding)
There exists a set S of volume at most δ whose expansion is at
most 2( )
Theorem (Approximating Isoperimetric Profile)
For every constant δ > 0, there exists a poly-time algorithm
that finds a set S of size O(δ) such that
expansion (S) ≤ O( ( ) log(1/  ) )
Restricted Eigenvalue Problem
Given a matrix A, find a submatrix A[S,S] of
size at most δn X δn matrix with the least
eigenvalue.
-- Our algorithm is applicable to diagonally
dominant matrices
(yields a log(1/δ) approximation).
Approximating Spectral Profile
( )  minn
2
(
x

x
)
 i j
( i , j )E
x
|Support(x)| < δn
d  xi
i
2
SDP Relaxation for
 ( )  min
n
2
(
x

x
)
 i j
( i , j )E
d  xi
x
|Support(x)| < δn
i
Replace each xi by a vector vi:
Num erator
2
|
v

v
|
 i j
Denominator 
| v |
( i , j )E
( i , j )E
2
i
=n
Without loss of generality, xi can be assumed positive.
This yields the constraint:
vi∙vj ≥ 0
Enforcing Sparsity:
2
By Cauchy-Schwartz inequality:
 vi
i


2
  xi  | Support( x) |  xi
i
 i 
2
 n  | vi |2  n 2
i
2
SDP Relaxation for
 ( )  min
n
2
(
x

x
)
 i j
( i , j )E
x
|Support(x)| < δn
Minimize (Sum of Squared Edge Lengths)
d  xi
i
2
|
v

v
|
 i j
( i , j )E
Subject to
Positive Inner Products:
Average squared length
vi∙vj ≥ 0
=1
Average Pairwise Correlation < δ
2
|
v
|
 i n
( i , j )E
2
v

v


n
 i j
i, j
2
Rounding
Two Phase Rounding:
• Transform SDP vectors in to a SDP solution with only nonnegative coordinates.
• Use thresholding to convert non-negative vectors in to
sparse vectors.
Making SDP solution nonnegative
Let v be a n-dimensional real vector.
Let v* denote the unit vector along direction v.
Map the vector v to the following function over Rn :
fv = ||v||∙ (Square Root of Probability Density Function
of n-dimensional Gaussian centered at v*)
Formally,
f v ( x)  v   ( x  v* )
Where:
Ф(x) = probability density function of a
mean 0, variance σ
spherical Gaussian in n-dimensions.
Properties
f v ( x)  v   ( x  v* )
Where:
Ф(x) = probability density
function of a mean 0, variance σ
spherical Gaussian in n-dimensions.
SDP Constraint
Average Pairwise Correlation < δ
2
v

v


n
 i j
i, j
Lemma: (Pairwise correlation remains low if σ is small)
2


2
 |  f i |   (e 1/ 4   )n 2


(
i
,
j
)

E


Pick σ = 1/sqrt{log(1/δ)}
Properties
f v ( x)  v   ( x  v* )
Where:
Ф(x) = probability density
function of a mean 0, variance σ
spherical Gaussian in n-dimensions.
Lemma: (Squared Distances get stretched by at most 1/σ2)
|f1 – f2 |2 ≤ O(1/σ2) |v1 – v2 |2
For our choice of σ, squared distances are stretched
by log(1/δ) .
With a log(1/δ) factor loss, we obtaining a non-negative
SDP solution.
Rounding a positive vector solution
Let us pretend the vectors vi are non-negative.
i.e.,
vi(t) ≥ 0
for all t
Rounding Non-Negative Vectors
1. Sample t
2. Compute threshold
θ = (average of vi(t) over i) * (2/ δ )
3. Set xi = max{ vi(t) – θ, 0 } for all i
Observation: |Support(x)| < δ/2 ∙ n
Observation: E[
 (x  x )
( i , j )E
i
j
2
]  vi  v j
2
Open Problem
Minimize (Sum of Squared Edge Lengths)
| v
( i , j )E
Subject to
Positive Inner Products:
Average squared length
i
 v j |2
vi∙vj ≥ 0
=1
Average Pairwise Correlation < δ
| v
( i , j )E
v
i
i
|2  n
 v j  n 2
i, j
Find integrality gaps for the SDP relaxation:
Current best have δ = 1/poly(logn),
Do there exist integrality gaps with δ = 1/poly(n)?
Thank You
Spectral Profile
Second
Eigen
Value:
Spectral
Profile
2
(
x

x
)
 i j

( ) min ( i , j )E
x n
|Support(x)| < δn
d  xi
2
xT Lx
 min
T
x n
dx x
i
Remarks:
• Λ(δ) ≤ minimum of Ф(δ0) over all δ0 ≤ δ
• Unlike second eigen value, Λ(δ) is not the optimum of a
convex program.
• Λ(δ) is the smallest possible eigenvalue of a submatrix of size
at most < δn of the matrix L
• xi can be assumed to be all positive.
Small Sets via Spectral Profile
Using an analysis along the lines of analysis of Cheeger’s
inequality, it yields:
Theorem [Raghavendra-Steurer-Tetali 10]
There exists a poly-time algorithm that finds a set S of
size O(δ) such that
expansion (S) ≤ sqrt (Λ* (δ)∙O(log 1/δ))
So if there is a set S with expansion(S) = ε, then
the algorithm finds a cut of size
O(  log1/  )
Similar behaviour as the Gaussian expansion profile
Spectral Profile
Second
Eigen
Value:
Spectral
Profile
2
(
x

x
)
 i j

( ) min ( i , j )E
x n
|Support(x)| < δn
d  xi
i
Replace xi by vectors vi
* ( )  min
vi
xT Lx
 min
T
x n
dx x
2
2
|
v

v
|
 i j
( i , j )E
d  | vi |2
i
Subject to
v
i
i
2
 n | vi |2
i
vi∙vj ≥ 0
Graph Expansion
d-regular graph G
d
vertex set S
Approximation Algorithms:
expansion(S) =
# edges leaving S
d |S|
A random
neighbor of a random
vertexGin S
Conductance
of Graph
is outside of S with probability expansion(S)
ФG =
minimum
expansion(S)
|S| ≤ n/2
Extremely well-studied, many different
Uniform Sparsest Cut Problem
contexts
Given a graph G compute
ФG
pseudo-randomness,
group theory,
androuting,
find the set S achieving it.
online
Markov chains, metric embeddings, …
•Cheeger’s Inequality [Alon][Alon-Milman]
Given a graph G, if the normalized adjacency matrix has second eigen value λ2 then,
(1  2 )
  G  2(1  2 )
2
•A log n approximation algorithm [Leighton-Rao].
•A sqrt(log n) approximation algorithm using semidefinite programming [Arora-Rao-Vazirani].
Limitations of Eigenvalues
• The best lower bound that Cheeger’s inequality gives on
expansion is (1-λ2)/2 < ½, while Ф(δ) can be close to 1.
• Consider graph G
Connect pairs of points on {0,1}n that are εn Hamming
distance away.
Then Second eigenvalue ≈ 1- ε and ф(1/2) ≈ ε
yet Ф(δ) ≈ 1 (small sets have near-perfect expansion)
• A SIMPLE SDP RELAXATION cannot distinguish between
- all small sets expand almost completely
- exists small set with almost no expansion
A Conjecture
Small-Set Expansion Conjecture:
8η>0, 9 ± >0 such that GapSSE(η, ± ) is NP-hard, i.e.,
Given a graph G, it is NP-hard to distinguish
YES: expansion(S) < η for some S of volume ¼ ±
NO: expansion(S) > 1- η for all S of volume ¼ ±
Road Map
• Algorithm:
– Spectral Profile.
• Reductions within Expansion
• Relationship with Unique Games Conjecture
Gaussian Curve
Gaussian Graph
Vertices: all points in Rd (d
dimensional real space)
Edges: Weights according to the
following sampling procedure:
• Sample a random Gaussian
variable x in Rd
• Perturb x as follows to get y in
Rd
y  (1   ) x  2   2  z
Γε (δ) = Gaussian noise
sensitivity of a set of
measure δ
= least expanding sets are
caps/thresholds of
measure δ
1
=
(  log1/  )
Ф(δ)
Add an edge between x an y
Set Size δ
0.5
Approximating Expansion
Profile
Reductions within Expansion
Reductions within Expansion
Theorem [Raghavendra-Steurer-Tulsiani 10]
For every positive integer q, and constants ε,δ,γ,
given a graph it is SSE-hard to distinguish between:
• There exists q disjoint small sets S1 , S2 , .. Sq of size close to
1/q, such that expansion(Si) ≤ ε
• No set of size μ> δ has expansion less than size Γε/2 (μ) -expansion of set of size μ in Gaussian graph with parameter
ε/2
expansion (S) < sqrt (Λ* (δ)∙O(log 1/ μ))
Informal Statement
1
1
Ф(δ)
0.5
Set Size δ
Set Size δ
Quantitative Statement
Qualitative
Assumption
Given a graph G , it is SSE-hard to distinguish whether,
•There is a small set of size whose expansion is ε.
GapSSE is NPhard
•Every small set of size μ (in a certain range) expands at
least as much as the corresponding Gaussian graph with
noise ε
O(  log1/  )
Corollaries
Corollary: The algorithm in [Raghavendra-Steurer Tetali 10] has
near-optimal guarantee assuming SSE conjecture.
Corollary: Assuming GapSSE, there is no constant factor
approximation for Balanced Separator or Uniform Sparsest
Cut.
Relation with Unique Games
Conjecture
Unique Games
Unique game ¡ :
label set
A
L(A)
Referee
graph
of size n
¼
B
Each edge (A,B) has an associated map ¼
¼ is bijection from L(A) to L(B)
A labelling satisfies (A,B) if
¼ (label of A) = label of B
Goal: Find an assignment of labels
to vertices such that maximum
number of edges are satisfied.
sample (A,B,¼)
A
B
Player 1
Player 2
pick a in L(A)
pick b in L(B)
a
b
Referee
players win
if ¼(a) = b
value( ¡ ): no communication
between
players
maximum success
probability
over all strategies of the players
Unique Games Conjecture [Khot02]
Unique Games Conjecture: [Khot ‘02]
8²>0, 9 q >0:
NP-hard to distinguish for ¡ with label set size q
YES: value( ¡ ) > 1-²
NO: value( ¡ ) < ²
Implications of UGC
BASIC SDP is optimal for …
UGC
Constraint Satisfaction Problems [Raghavendra`08]
MAX CUT, MAX 2SAT
Metric Labeling Problems [MNRS`08]
MULTIWAY CUT, 0-EXTENSION
Ordering CSPs [GMR`08]
MAX ACYCLIC SUBGRAPH, BETWEENESS
Strict Monotone CSPs [KMTV`10]
VERTEX COVER, HYPERGRAPH VERTEX COVER
Kernel Clustering Problems [KN`08,10]
Grothendieck Problems [KNS`08, RS`09]
…
“Reverse Reductions”
BASIC SDP optimal
for PROBLEM X
?
*
UGC
BASIC SDP is optimal for
lots of optimization problems,
e.g.: MAX CUT and VERTEX COVER
Win-Win Situation
If we could show * , then a refutation of UGC would
imply an improved algorithm for PROBLEM X
PROBLEM X = MAX CUT
Parallel Repetition is natural candidate reduction for * [FeigeKinderO’Donnell’07]
Bad news: this reduction cannot work [Raz’08, BHHRRS’08]
Small Set Expansion and Unique
Games
• Solving Unique Games  Finding a small nonexpanding set in the “label extended graph”
Theorem [Raghavendra-Steurer 10]
Small Set Expansion Conjecture  Unique Games Conjecture
Establishes a reverse connection from a
natural problem.
Implications of UGC
BASIC SDP is optimal for …
UGC
UGC
With expansion
Gap SSE
Constraint Satisfaction Problems [Raghavendra`08]
MAX CUT, MAX 2SAT
Metric Labeling Problems [MNRS`08]
MULTIWAY CUT, 0-EXTENSION
Ordering CSPs [GMR`08]
MAX ACYCLIC SUBGRAPH, BETWEENESS
Strict Monotone CSPs [KMTV`10]
VERTEX COVER, HYPERGRAPH VERTEX COVER
Kernel Clustering Problems [KN`08,10]
Grothendieck Problems [KNS`08, RS`09]
…
Uniform Sparsest Cut [KNS`08, RS`09]
Minimum Linear Arrangement[KNS`08, RS`09]
Most known SDP integrality gap instances for problems
like MaxCut, Vertex Cover, Unique games have graphs
that are “small set expanders”
Theorem [Raghavendra-Steurer-Tulsiani 10]
Small Set Expansion Conjecture  MaxCut or Unique Games
on Small Set Expanders is hard.
Reverse Connections?
Approximating Spectral Profile
Roadmap
Introduction
Graph Expansion: Cheeger’s Inequality, Leighton Rao, ARV
Expansion Profile: Small Sets expand more than large ones.
Cheeger’s inequality and SDPs fail
GapSSE Problem
Relation to Unique Games Conjecture: Unique Games definition,
Applications, lack of reverse reductions. Label extended graph  Small sets
Small Set Expansion Conjecture  UGC
UGC with SSE is easy
Algorithm for SSE: Spectral Profile, SDP for Spectral Profile, Rounding
algorithm
Relations within expansion:
GapSSE  Balanced Separator Hardness
NP-hard Optimization
Example:
MAX CUT: partition vertices of a graph into two sets
so as to maximize number of cut edges
fundamental graph partitioning problem
benchmark for algorithmic techniques
Approximation
MAX CUT
Trivial approximation
Random assignment, cut ½ of the edges
First non-trivial approximation
approx-ratio 0.878
Goemans–Williamson algorithm (‘95)
based on a semidefinite relaxation (BASIC SDP)
Beyond Max Cut:
Analogous BASIC SDP relaxation for many other problems
Almost always, BASIC SDP gives best known approximation
(often strictly better than non-SDP methods)
Approximation
MAX CUT
Trivial approximation
Random assignment, cut ½ of the edges
First non-trivial approximation
approx-ratio 0.878
Goemans–Williamson algorithm (‘95)
based on a semidefinite relaxation (BASIC SDP)
Can we beat the approximation guarantee of BASIC SDP?
Unique Games Conjecture
Can we beat the approximation guarantee of BASIC SDP for Max Cut?
No, assuming Khot’s Unique Games Conjecture!
[KKMO`05, MOO`05, OW’08]
Unique Games Conjecture [Khot’02] (roughly):
it is NP-hard to approximate the value of Unique Games
certain optimization problem:
given equations of the form
xi – xj = cij mod q
satisfy as many as possible
UGC: Is it true?
What are hard instances?
Any non-trivial consequences
if UGC is false?
UGC
for certain SDP
hierarchies
UGC
?
[RaghavendraS’09,KhotSaket’09]
UGC
on expanding
instances
[AKKSTV’08,
AIMS’10]
UGC
on product
instances
[BHHRRS’08,Raz’08]
“Reverse Reductions”
BASIC SDP optimal
for PROBLEM X
UGC
BASIC SDP is optimal for
lots of optimization problems,
e.g.: MAX CUT and VERTEX COVER
SMALL-SET EXPANSION
Win-Win Situation
A refutation of UGC implies an improved algorithm
for SMALL-SET EXPANSION (better than BASIC SDP!)
First reduction from natural combinatorial problem to UNIQUE GAMES
Approximating Small-Set Expansion
Expansion profile of G at ±:
minimum expansion(S) over all S with volume < ±
How well can we approximate the
expansion profile of G for small ± ?
BASIC SDP cannot distinguish between
- all small sets expand almost completely
- exists small set with almost no expansion
Small-Set Expansion Conjecture:
8²>0, 9 ± >0:
NP-hard to distinguish
YES: expansion(S) < ² for some S of volume ¼ ±
NO: expansion(S) > 1-² for all S of volume ¼ ±
S
Unique 2-Prover Games
Unique game ¡ :
label set
A
L(A)
Referee
sample (A,B,¼) from D
universe
of size n
¼
B
A
label set
L(B)
distribution D over triples (A,B,¼)
- A and B are from U
- ¼ is bijection from L(A) to L(B)
value( ¡ ):
maximum success probability
over all strategies of the players
B
Player 1
Player 2
pick a in L(A)
pick b in L(B)
a
b
Referee
players win
if ¼(a) = b
no communication
between players
Approximating Unique Games
How well can we approximate the value of a unique game
for large label sets?
BASIC SDP cannot distinguish between games with value ¼ 1 and ¼ 0
for large label sets
Unique Games Conjecture: [Khot ‘02]
8²>0, 9 q >0:
NP-hard to distinguish for ¡ with label set size q
YES: value( ¡ ) > 1-²
NO: value( ¡ ) < ²
Approximating Unique Games
Unique Games Conjecture:
8²>0, 9 q >0:
NP-hard to distinguish for ¡ with label set size q
YES: value( ¡ ) > 1-²
NO: value( ¡ ) < ²
Our main theorem:
Small-Set Expansion Conjecture ) Unique Games Conjecture
Reduction:
Small-Set Expansion  Unique Games
Task:
find non-expanding set of volume ¼ ±
graph G
B
A
Referee
sample R = 1/± random edges M
A = one half of each edge
B = other half of each edge
A
B
Player 1
Player 2
pick a 2 A
pick b 2 B
a
b
Referee
players win
if (a,b) 2 M
Completeness
Small Non-Expanding Set  (Partial) Strategy
graph G
S
referee allows
players to refuse
for few queries
B
A
Suppose expansion(S) < ²
With constant probability, |A Å S |= {a}
Conditioned on this event:
P(
other half of a’s edge
outside of S
)<²
 partial game value > 1 - ²
Strategy for Player 1:
pick a 2 A if a is unique intersection with S
otherwise, refuse to answer
Soundness
Strategy  Small Non-Expanding Set
standard trick:
can assume both players have same strategy
Idea: strategy  distribution over sets
- sample R-1 vertices U
- output S = { x | Player 1 picks x if A=U+x }
Easy to show:
E volume(S) = 1/R = ±
E # edges leaving S
Suppose:
players win with prob > 1-²
E d |S|
<²
Are we done?
No!
Problem:
volume(S) might not be concentrated around ±
Soundness
Strategy  Small Non-Expanding Set
² R random vertices
+ ²-noise
+ ²-noise
Idea: Referee adds ²-noise to A and B
New distribution over sets:
- sample R-1 vertices U
- S = { x | players pick x if A = U+x+noise
with probability > ½ }
Can show:
Suppose:
players win with prob > 1-²
1
8 U. volume(S) < # noise vertices
=±/²
Intuition:
players cannot distinguish x and noise
Summary
BASIC SDP optimal
for SMALL-SET
EXPANSION
UGC
BASIC SDP is optimal for
lots of optimization problems,
e.g.: MAX CUT and VERTEX COVER
Open Questions (+ Subsequent Work)
more reverse reductions?
noise here vs. noise in other hardness reductions?
Hardness results based on Small-Set Expansion Conjecture?
 Reductions between Expansion Problems [Raghavendra S Tulsiani’10]
Better algorithms for UNIQUE GAMES via SMALL-SET EXPANSION?
poly(²)
 2n
algorithm for (1-², ½)-UG via SSE [Arora Barak S’10]
Thanks!
Questions?
APPENDIX
Rounding a UG strategy
to a small non-expanding set
Unique Games
Unique Games instance ¡ :
random edge (u,v)
decorated
bipartite graph
u
u
v
label set L(u)
Referee
label set L(v)
bijection ¼uv:
L(u)  L(v)
value( ¡ ):
maximum success probability
over all strategies of the players
v
Player 1
Player 2
pick i in L(u)
pick j in L(v)
i
j
Referee
players win
if ¼uv(i) = j
each player knows only
half of referee’s edge
Approximation
MAX CUT
Trivial approximation
Random assignment, cut ½ of the edges
First non-trivial approximation
approx-ratio 0.878
Goemans–Williamson algorithm (‘95)
based on a semidefinite relaxation (BASIC SDP)
General constraint satisfaction problems (CSPs):
simple generic approximation algorithm [RaghavendraS’09]
for every CSP: approximation matches integrality gap of certain SDP
near-linear running time [S’10]
Can we beat the approximation guarantee of BASIC SDP (for Max Cut)?
Combinatorial Optimization
Example:
MAX CUT: partition vertices of a graph into two sets
so as to maximize number of cut edges
A concrete practical application:
¼ MAX CUT
CAIDA at UCSD analyzes business relationships
among Autonomous Systems in the internet using MAX 2SAT