Venkatesan Guruswami (CMU)
Yury Makarychev (TTI-C)
Prasad Raghavendra (Georgia Tech)
David Steurer (MSR)
Yuan Zhou (CMU)
MaxCut and Goemans-Williamson alg.
G = (V, E)
A
Objective:
 edges( A, B) 
max

|E|


B
• The GW SDP relaxation [GW95]
1  vi , v j
1
max

| E | (i , j )E
2
subject
to
– 0.878-approximation
– (1   ) vs (1   ) approximation
|| vi ||2  1, i V
Finding almost-perfect MaxCut
• (1   ) vs (1   ) approximation
– Bipartite graph recognition algorithm (robust version
against noise)
– Optimal under Unique Games Conjecture [KKMO07,
MOO10]
MaxBisection
G = (V, E)
A
Objective:
B
| V | even
 edges( A, B) 
max

|E|


| A || B |
• Approximating MaxBisection?
– No easier than MaxCut
• Reduction: take two copies of the MaxCut instance
MaxBisection (cont'd)
G = (V, E)
A
Objective:
B
| V | even
 edges( A, B) 
max

|E|


| A || B |
• Approximating MaxBisection?
– No easier than MaxCut
– Strictly harder than MaxCut?
– Approximation ratio: 0.7028 [FJ97, Ye01, HZ02, FL06]
– Approximating almost perfect solutions? Not known
Finding almost-perfect MaxBisection
• Question
– Is there a (1   ) vs (1  o (1)) approximation
algorithm for MaxBisection?
• Answer. Yes.
• Our result.
– Theorem. There is a (1   ) vs (1   1/ 3 log 1 )
approximation algorithm for MaxBisection.
– Theorem. Given a (1   ) satisfiable MaxBisection
instance, it is easy to find a (.49, .51)-balanced cut of
value (1   ) .
The rest of this talk...
• Theorem. There is a (1   ) vs (1   1/ 20 log n)
approximation algorithm for MaxBisection.
Approach
Approach -- SDP
• The standard SDP (used by all the previous algorithms)
1  vi , v j
1
max

| E | (i , j )E
2
• Integrality gap
OPT < 0.9
, subject to
|| vi ||2  1, i V
v
iV
SDP = 1
i
0
Our approach
A simple fact
• Fact.(1 / 2   ,1 / 2   ) -balanced cut of value c
of value c  2 .
 bisection
• Proof. Get the bisection by moving  fraction of
random vertices from the right side to the left side.
• Only need to find almost bisections.
Almost perfect MaxCuts on expanders
• λ-expander: for each S  V, such that vol( S )  vol(V ) / 2 ,
we have edges( S , V  S )
, where vol( S )   di

iS
vol( S )
• Key Observation. The (volume of) difference between
two (1   ) cuts on a λ-expander is at most 2 /   vol(V ) .
• Proof.
cut( A, B)  1   cut(C , D)  1  
C
X
edges( X  Y ,V  X  Y )  2  vol(V )


A
B
vol( X  Y )  2 /   vol(V )
 
Y
D
Almost perfect MaxCuts on expanders
• λ-expander: for each S  V, such that vol( S )  vol(V ) / 2 ,
we have edges( S , V  S )
, where vol( S )   di

iS
vol( S )
• Key Observation. The (volume of) difference between
two (1   ) cuts on a λ-expander is at most 2 /   vol(V ) .
• Approximating almost perfect MaxBisection on
expanders is easy.
– Just run the GW alg. to find the MaxCut.
The algorithm (sketch)
• Decompose the graph into expanders
– Discard all the inter-expander edges
• Approximate OPT's behavior on each expander by
finding MaxCut (GW)
– Discard all the uncut edges
• Combine the cuts on the expanders
– Take one side from each cut to get an almost
bisection. (subset sum)
Expander decomposition
• Cheeger's inequality. Can (efficiently) find a cut of
sparsity  if the graph is not a  -expander.
• Corollary. A graph can be (efficiently) decomposed into
 -expanders by removing  log n edges (in fraction).
• Proof.
– If the graph is not an expander, divide it into two
parts by sparsest cut (cheeger's inequality).
– Process the two parts recursively.
The algorithm
• Decompose the graph into
1/ 20
log n edges.
– Lose 
 1/10-expanders.
• Apply GW algorithm on each expander to approximate OPT.
– OPT(MaxBisection) = (1   )
– GW finds (1   ) cuts on these expanders
1/ 2
1/ 10
  1/10 different from behavior of OPT
•  /
– Lose  1 / 2 edges.
• Combine the cuts on the expanders (subset sum).
•
•
( 2 
1
1/10 1
,
2


1/ 20
-balanced
cut
of
value
(
1


log n)
)
a bisection of value (1   1/ 20 log n)
1/10
Eliminating the log n factor
• Another key step.
• Idea. Terminate early in the decomposition process.
Decompose the graph into  1/10-expanders or subgraphs
of   n vertices.
• Corollary. Only need to discard  1/ 20 log 1 edges.
• Lemma. We can find an almost bisection if the MaxCuts
for small sets are more biased than those in OPT.
Finding a biased MaxCut
• Lemma. Given G=(V,E), if there exists a cut (X, Y) of
value (1   ) , then one can find a cut (A, B) of
value (1   ) , such that | A || X |   | V .|
• SDP.
1
maximize
subject to
v ,v

|V |
iV
0
i
1  vi , v j
1
 1 

| E | ( i , j )V
2
|| vi ||2  1
2
2
 -triangle
inequality
| vi  v j , v0 |
, i {0} V
|| vi  v j ||2
2
, i, j V
• Rounding. A hybrid of hyperplane and threshold rounding.
Future directions
• (1   ) vs (1   )
approximation?
• "Global conditions" for other CSPs.
– Balanced Unique Games?
The End.
Any questions?