Venkatesan Guruswami (CMU) Yury Makarychev (TTI-C) Prasad Raghavendra (Georgia Tech) David Steurer (MSR) Yuan Zhou (CMU)

Venkatesan Guruswami (CMU)Yury Makarychev (TTI-C)

Prasad Raghavendra (Georgia Tech)David Steurer (MSR)

Yuan Zhou (CMU)

Bipartite graph recognition• Depth-first search/breadth-first search

• With some noise?– Given a bipartite graph with 1% noisy edges, can we remove a small fraction of edges (10% say) to

get a bipartite graph, i.e. can we divide the vertices into two parts, so that 9

0% of the edges go accross the two parts?

MaxCut• cut(A, B) = edges(A, B) / |E| where B = V - A• exact one of i, j in A : edge (i, j) "on the cut"• MaxCut: find A, B such that cut(A, B) is maximized

– Bipartite graph recognition: MaxCut = 1 ?– Robust bipartite graph recognition: given MaxCut ≥ 0.

99, to find cut(A, B) ≥ 0.9

G=(V,E)

A B = V - A

cut(A, B) = 4/5

Eji

ji xx

EMaxCut

),( 2

1

||

1max subject

toVixi },1,1{ Vixi ,12

c vs. s approximation for MaxCut• Given a graph with MaxCut value at least c, can we find a c

ut of value at least s ?

• Robust bipartite graph recognition: given MaxCut ≥ 0.99, to find cut(A, B) ≥ 0.9– 0.99 vs 0.9 approximation – "approximating almost perfect MaxCut"

Robust bipartite graph recognition• Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9

• We can always find cut(A, B) ≥ 1/2.– Assign each vertex -1, 1 randomly– For any edge (i, j), E[(1 - xixj)/2] = 1/2

vi vj

2

1

2

1E

||

1

2

1

||

1E]alg[E

),(

),(

Eji

ji

Eji

ji

xx

E

xx

E

Robust bipartite graph recognition (cont'd)

• Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9

• We can always find cut(A, B) ≥ 1/2.• Better than 1/2?

– DFS/BFS/greedy?

– Linear Programming?

No combinatorial algorithm known until very recent [KS11]

Natural LPs have big Integrality Gaps [VK07, STT07, CMM09]

Robust bipartite graph recognition (cont'd)

• Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9

• We can always find cut(A, B) ≥ 1/2.• Better than 1/2?

• The GW Semidefinite Programming relaxation [GW95]

– 0.878-approximation– Given MaxCut , can find a cut

• vs approximation, tight under Unique Games Conjecture [Kho02, KKMO07, MOO10]

Eji

ji vv

E ),( 2

,1

||

1max

)1( )1(

subject to

Vivi ,1|||| 2

)1( )1(

Robust satisfiability algorithms• Given an instance which can be satisfied by removing ε fr

action of constraints, to make the instance satisfiable by removing g(ε) fraction of constraints – g(ε) -> 0 as ε -> 0

• Examples• vs. algorithm for MaxCut [GW95]• vs. algorithm for Max2SAT [Zwick98]• vs. algorithm for MaxHorn3SAT [Zwick98]

)1( )1( )1( )1( )1( )log/11( 1

MaxBisection

G = (V, E)

A B

Objective:

|||| BA evenV ||

Eji

ji xx

EonMaxBisecti

),( 2

1

||

1max

Vixi ,12

subject to 0

Viix

||

),(),(max

E

BAedgesBAcut

MaxBisection (cont'd)

• Approximating MaxBisection?– No easier than MaxCut

•Reduction: take two copies of the MaxCut instance

||

),(),(max

E

BAedgesBAcut

G = (V, E)

A B

Objective:

|||| BA evenV ||

MaxBisection (cont'd)

• Approximating MaxBisection?– No easier than MaxCut– Strictly harder than MaxCut?– Approximation ratio: 0.6514 [FJ97], 0.699 [Ye01], 0.7016

[HZ02], 0.7027 [FL06]– Approximating almost perfect solutions? Not known

G = (V, E)

AA BB

Objective:

|||| BA evenV ||

||

),(),(max

E

BAedgesBAcut

Finding almost-perfect MaxBisection• Question

– Is there a vs approximation algorithm for MaxBisection, where ?

• Answer. Yes.

• Our result.– Theorem. There is a vs approximation alg

orithm for MaxBisection. – Theorem. Given a satisfiable MaxBisection instanc

e, it is easy to find a (.49, .51)-balanced cut of value .

)1( ))(1( g

)1( )log1( 13/1

)1(

)1(

00)( asg

Extension to MinBisection• MinBisection

– minimize edges(A, B)/|V|, s.t. B = V - A, |B| = |A|

• Our result– Theorem. There is a vs approximation algor

ithm for MaxBisection. – Theorem. Given a MinBisection instance of value , it

is easy to find a (.49, .51)-balanced cut of value .

13/1 log

The rest of this talk...

• Previous algorithms for MaxBisection.

• Theorem. There is a vs approximation algorithm for MaxBisection.

)1( )log1( 20/1 n

Previous algorithms for MaxBisection

The GW algorithm for (almost perfect) MaxCut [GW95]

• MaxCut objective

• SDP relaxation

Eji

ji xx

EMaxCut

),( 2

1

||

1max subject

toVixi ,12

-1 0 1

subject to

Vivi ,1|||| 2

MaxCut = 2/3

SDP ≥ MaxCut

Eji

ji vv

ESDP

),( 2

,1

||

1max

In this example:SDP = 3/4 > MaxCut

The "rounding" algorithm

• Lemma. We can (in poly time) get a cut of value when• Algorithm. Choose a random hyperplane, the hyperplan

e divides the vertices into two parts. • Analysis.

11SDP

subject to

Vivi ,1|||| 2

Eji

ji vv

ESDP

),( 2

,1

||

1max

The "rounding" algorithm (cont'd)

• Lemma. We can (in poly time) get a cut of value when• Algorithm. Choose a random hyperplane, the hyperplan

e divides the vertices into two parts. • Analysis.

– implies for most edges (i, j), their SDP contribution is large

– Claim. If , then

– Therefore, the random hyperplane cuts many edges (in expectation)

11SDP

subject to

Vivi ,1|||| 2

Eji

ji vv

ESDP

),( 2

,1

||

1max

12/),1( ji vv1]hyperplanerandombyseperated,Pr[ ji vv

1SDP2/),1( ji vv

The "rounding" algorithm (cont'd)

– Claim. If , then

– Proof.

12/),1( ji vv1]hyperplanerandombyseperated,Pr[ ji vv

vi vj

vi, vj seperated by the hyperplane

vi, vj not seperated by the hyperplane

O

vv

vv

vv

ji

ji

ji

121arccos

1

,arccos1

2

,arccos2

2

2

]seperated,Pr[

21, ji vv

Known algorithms for MaxBisection• The standard SDP (used by all the previous algorithms)

• Gives non-trivial approximation gaurantee

• But does not help find almost perfect MaxBisection

Eji

ji vv

E ),( 2

,1

||

1max

0Vi

iv, subject to

Vivi ,1|||| 2

Bisection condition

Known algorithms for MaxBisection (cont'd)

• The standard SDP (used by all the previous algorithms)

• The "integrality gap"

Eji

ji vv

E ),( 2

,1

||

1max

0Vi

iv, subject to

OPT < 0.9 SDP = 1

Vivi ,1|||| 2

Known algorithms for MaxBisection (cont'd)

• The standard SDP (used by all the previous algorithms)

• The "integrality gap" : instances that OPT < 0.9, SDP = 1

• Why is this a bad news for SDP?– Instances that OPT > 1 - ε, SDP > 1 - ε– Instances that OPT < 0.9, SDP > 1 - ε

– SDP cannot tell whether an instance is almost satisfiable (OPT > 1 - ε) or not.

Eji

ji vv

E ),( 2

,1

||

1max

0Vi

iv, subject to

Vivi ,1|||| 2

Our approach

• Theorem. There is a vs approximation algorithm for MaxBisection.

)1( )log1( 20/1 n

A simple fact• Fact. -balanced cut of value bisection of val

ue .

• Proof. Get the bisection by moving fraction of random vertices from the large side to the small side.– fraction of cut edges affected : at most in expectati

on

• Only need to find almost bisections.

)2/1,2/1( c2c

2

Almost perfect MaxCuts on expanders• λ-expander: for each , such that , we have , where

)(

),(

Svol

SVSedgesVS 2/)()( VvolSvol

Si

idSvol )(

G=(V,E)G=(V,E)

SS

Almost perfect MaxCuts on expanders (cont'd)

• λ-expander: for each , such that , we have , where

• Key Observation. The (volume of) difference between two cuts on a λ-expander is at most .

• Proof.

)(

),(

Svol


)1( )(/2 Vvol

AA BB

CC

DD

1),( BAcut 1),( DCcut

XX

YY

)(2),( VvolYXVYXedges

)(/2)( VvolYXvol

Si

idSvol )(

Almost perfect MaxCuts on expanders (cont'd)

• λ-expander: for each , such that , we have , where

• Key Observation. The (volume of) difference between two cuts on a λ-expander is at most .

• Approximating almost perfect MaxBisection on expanders is easy.– Just run the GW alg. to find the MaxCut.

)(

),(

Svol


)1( )(/2 Vvol

Si

idSvol )(

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)

G=(V,E)G=(V,E)

Step 1: decompose into expanders

Step 2: find MaxCutStep 3:combine pieces

Expander decomposition• Cheeger's inequality. Can (efficiently) find a cut of sparsi

ty if the graph is not a -expander.• Corollary. A graph can be (efficiently) decomposed into

-expanders by removing edges (in fraction).• Proof.

– If the graph is not an expander, divide it into small parts by sparsest cut (cheeger's inequality).

– Process the small parts recursively.

nlog

G=(V,E)G=(V,E)

λ-expanderλ-expander

The algorithm • Decompose the graph into -expanders.

– Lose edges.

• Apply GW algorithm on each expander to approximate OPT.– OPT(MaxBisection) = – GW finds cuts on these expanders

• different from behavior of OPT– Lose edges.

• Combine the cuts on the expanders (subset sum).

• -balanced cut of value • a bisection of value

10/1

)1( )1(

10/110/12/1 /

nlog20/1

2/1

),( 10/12

110/12

1 )log1( 20/1 n)log1( 20/1 n

• Proved:• Theorem. There is a vs approximation al

gorithm for MaxBisection.

• Will prove:• Theorem. There is a vs approximation al

gorithm for MaxBisection.

)1( )log1( 20/1 n

)1( )log1( 120/1

short story

Eliminating the factor• Recall. Only need to find almost bisections ( -close to a

bisection)

• Observation. Subset sum is "flexible with small items"– Making small items more biased does not change the solution too much.

nlog20/1

(101, 304)(397, 201)(3, 5)(6, 2)(5, 1)(3, 2)

(515, 515)sum

(8, 0)(8, 0)(6, 0)(5, 0)


bisection)


nlog20/1

(101, 304)(397, 201)

(498, 505)sum

(8, 0)(8, 0)(6, 0)(5, 0)


bisection)


nlog20/1

(101, 304)(397, 201)

(506, 505)sum

(8, 0)(8, 0)(6, 0)(5, 0)


bisection)


nlog20/1

(101, 304)(397, 201)

(506, 513)sum

(8, 0)(0, 8)(6, 0)(5, 0)


bisection)


nlog20/1

(101, 304)(397, 201)

(512, 513)sum

(8, 0)(0, 8)(6, 0)(5, 0)


bisection)


nlog20/1

(101, 304)(397, 201)

(517, 513)sum

(8, 0)(0, 8)(6, 0)(5, 0)


bisection)

• Observation. Subset sum is "flexible with small items"– Making small items more biased does not change the solution too much.– However, making small items more balanced might be a bad idea.

nlog20/1

(200, 0)(0, 2)

(200, 200)sum

(0, 2)

(0, 2)

100copies


bisection)

• Observation. Subset sum is "flexible with small items"– Making small items more biased does not change the solution too much.– However, making small items more balanced might be a bad idea.

nlog20/1

(200, 0)(1, 1)

(300, 100)sum

(1, 1)

(1, 1)

100copies

Eliminating the factor (cont'd)

• Idea. Terminate early in the decomposition process. Decompose the graph into – -expanders (large items), or – subgraphs of vertices (small items).

• Corollary. Only need to discard edges.

• Lemma. We can find an almost bisection if the MaxCuts we get for small sets are more biased than those in OPT.

nlog

10/1n

120/1 log

Finding a biased MaxCut• To find a cut that is as biased as OPT and as good as OPT

(in terms of cut value).

• Lemma. Given G=(V,E), if there exists a cut (X, Y) of value , then one can find a cut (A, B) of value , such that .

)1( )1( |||||| VXA

MaxBisectionMaxBisection Biased MaxCutBiased MaxCut

The algorithm • Decompose the graph into -expanders or small parts.

– Lose edges.

• Apply GW algorithm on each expander to approximate OPT.– Lose edges, different from OPT

• Find biased MaxCuts in small parts.– Lose edges, at most less biased than OPT

• Combine the cuts on the expanders and small parts (subset sum).

• -balanced cut of value • a bisection of value

10/1

10/110/12/1 /

120/1 log

2/1

),( 10/12

110/12

1 )log1( 120/1

2/1

)log1( 120/1

2/1

Finding a biased MaxCut -- A simpler task• Lemma. Given G=(V,E), if there exists a cut (X, Y) of value

, then one can find a cut (A, B) of value , such that .

• SDP.

• Claim. SDP ≥ |X|/|V|

)1( )1( 8 |||||| 8 VXA

Vi

ivv

V 2

,1

||

1 0

maximize

subject to

Vji

ji vv

E ),(

12

,1

||

1

Vivi }0{,1|||| 2

--- Bias

--- Cut value

Rounding algorithm (sketch)• Goal: given SDP solution, to find a cut (A, B) such that

– –

• For most ( fraction) edges (i, j), we have

• vi, vj are almost opposite to each other: vi ≈ - vj,

• Indeed,

81),( BAcut8||/|| SDPVA

12

,1 ji vv

1

00 ,, vvvv ji

422

000

2,2

|,||,,|

jiji

jijiji

vvvv

vvvvvvvvv

21,1 ji vv

Rounding algorithm (sketch) (cont'd)

• Project all vectors to v0

• Divide v0 axis into intervals– length =

• Most ( fraction ) edges' incident vertices fall into opposite intervals (good edges)

• Discard all bad edges

400 ,, vvvv jifor most edges (i, j):

v0

48

81

I(-4) I(-3) I(-2) I(-1) I(1) I(2) I(3) I(4)

8


• Let the cut (A, B) be– for each pair of intervals I(k) and I(-k), let A include the one with more vertices, B include the other

• (A, B) cuts all good edges v0

-4 -3 -2 -1 1 2 3 481),( BAcut


• Let the cut (A, B) be– for each pair of intervals I(k) and I(-k), let A include the one with more vertices, B include the other

• For each i in I(k) For each i in I(-k)8

0 )1(, kvvi

8

)()(

0 |))(||)((|||})(||,)(max{|2

,1kIkIkIkI

vv

kIkIi

i

80 )1(, kvvi

8

)()(

0 ||||})(||,)(max{|2

,1VkIkI

vvSDP

kk kIkIi

i

8|||| VA

Finding a biased MaxCut • Lemma. Given G=(V,E), if there exists a cut (X, Y) of value


• SDP.

)1( )1( |||||| VXA

Vi

ivv

V 2

,1

||

1 0

maximize

subject to

Vji

ji vv

E ),(

12

,1

||

1

Vivi }0{,1|||| 2

Vjivv

vvv jiji

,,

2

|||||,|

2

0 -triangle inequality

22

--- Bias

--- Cut value

Future directions• vs approximation?

• "Global conditions" for other CSPs.– Balanced Unique Games?

)1( )1(

The End.

Any questions?

Eliminating the factor• Another key step.

• Idea. Terminate early in the decomposition process. Decompose the graph into -expanders or subgraphs of vertices.

• Corollary. Only need to discard edges.

• Lemma. We can find an almost bisection if the MaxCuts for small sets are more biased than those in OPT.

nlog

10/1n

120/1 log

MaxBisectionMaxBisection Biased MaxCutBiased MaxCut

Finding a biased MaxCut• Lemma. Given G=(V,E), if there exists a cut (X, Y) of value


• SDP.

• Rounding. A hybrid of hyperplane and threshold rounding.

)1( )1( |||||| VXA

Vi

ivvV

,||

10maximize

subject to

Vji

ji vv

E ),(

12

,1

||

1

Vivi }0{,1|||| 2

Vjivv

vvv jiji

,,

2

|||||,|

2

0 -triangle inequality

22

Future directions• vs approximation?

• "Global conditions" for other CSPs.– Balanced Unique Games?

)1( )1(

The End.

Any questions?

Documents

Venkatesan Guruswami (CMU) Yury Makarychev (TTI-C) Prasad Raghavendra (Georgia Tech) David Steurer (MSR) Yuan Zhou (CMU)