Boaz Barak (MSR New England)Fernando G.S.L. Brandão (Universidade Federal de Minas Gerais)

Aram W. Harrow (University of Washington)

Jonathan Kelner (MIT)David Steurer (MSR New England)

Yuan Zhou (CMU)

Motivation

• Unique Games Conjecture (UGC) [Khot'02]– For every constant ε, the following

problem is NP-Hard

– UG(ε) : given a set of linear modulo equations in the form

– Distinguish•YES case: at least (1-ε) of the equations

are satisfiable•NO case: at most ε of the equations are

satisfiable

)(mod qcyx ji

Motivation (cont'd)• Unique Games Conjecture [Khot'02]

• Implications of UGC– For large class of problems, BASIC-SDP

(semidefinite programming relaxation) achieves optimal approximation

Examples: Max-Cut, Vertex-Cover, any Max-CSP

[KKMO '07, MOO '10, KV '03, Rag '08]

Open question: Is UGC true?

Previous results

• Evidences in favor of UGC– (Certain) strong SDP relaxation hierarchies

cannot solve UG in polynomial time [RS'09, KPS'10, BGHMRS'11]

• Evidences opposing UGC– Subexponential time algorithms solve UG [ABS '10]

Our results• Evidences in favor of UGC

– (Certain) strong SDP relaxation hierarchies cannot solve UG in polynomial time [RS'09, KPS'10, BGHMRS'11]

• Evidences opposing UGC– Subexponential time algorithms solve UG [ABS '10]

1.New evidence: a natural SDP hierarchy solves all previous "hard instances" for UG

2.New (indirect) evidence: natural generalization of UG

•still has ABS-type subexp-time algorithm•cannot be solved in poly-time assuming ETH

More about our first result

1.New evidence: a natural SDP hierarchy solves all previous "hard instances" for UG

Semidefinite programming (SDP) relaxation hierarchies

r

…

? rounds SDP relaxation in roughly timee.g. [SA'90,LS'91]

)(rOn

UG(ε)

• For certain SDP relaxation hierarchies– Upper bound : rounds suffice [ABS'10,BRS'11,GS'11]

– Lower bound : rounds needed [RS'09,BGHMRS'11]

)1(n

))logexp((log )1(n

Semidefinite programming (SDP) relaxation hierarchies

…

?

UG(ε)

• Theorem.– Sum-of-Squares (SoS) hierarchy (a.k.a.

Lasserre hierarchy [Par'00, Las'01]) solves all known UG instances within 8 rounds

(cont'd)

r rounds SDP relaxation in roughly timee.g. [SA'90,LS'91]

)(rOn

Proof overview

• Integrality gap instance– SDP completeness: a good vector solution– Integral soundness: no good integral

solution

• A common method to construct gaps (e.g. [RS'09])

– Use the instance derived from a hardness reduction

– Lift the completeness proof to vector world– Use the soundness proof directly

Proof overview (cont'd)

• Our goal: to prove there is no good vector solution– Rounding algorithms?

• Instead, – we bound the value of the dual of the SDP– interpret the dual of the SDP as a proof

system– lift the soundness proof to the proof

system

Sum-of-Squares proof system

0)(

0)(

0)(

2

1

xp

xp

xp

m

Axioms Goal

0)( xqderive

• Rules– – – – all intermediate polynomials have bounded

degree d ("level-d" SoS proof system)

0)()(0)(,0)( 2121 xpxpxpxp0)()(0)(,0)( 2121 xpxpxpxp

0)( 2 xh

"Positivstellensatz" [Stengle'74]

Example

• SoS proof:

Axioms Goal02 xx 01 x

derive

0)1()(1 22 xxxx

Axiom squared term

Formulate UG as a polynomial optimization problem, and re-write the existing

soundness proof in a SoS fashion (as above) !

Components of the soundness proof

• "Easy" parts– Cauchy-Schwarz/Hölder's inequality– Influence decoding

• "Hard" parts– Hypercontractivity inequality– Invariance Principle

(of known UG instances)

SoS proof of hypercontractivity• The 2->4 hypercontractivity inequality: for low degree polynomial

we have

• The goal of an SoS proof is to show

is sum of squares

• Prove by induction (very similar to the well-known inductive proof of the inequality itself)...

dSnS

iSi

S xxf||],[

)(

2/12

}1,1{

4/14

}1,1{])([3])([

xfExfE

nn x

d

x

])([])([9 4

}1,1{

22

}1,1{xfExfE

nn xx

d

Components of the soundness proof• "Easy" parts

– Cauchy-Schwarz/Hölder's inequality– Influence decoding– Independent rounding

• "Hard" parts– Hypercontractivity inequality– Invariance Principle

•trickier •"bump function" is used in the original proof --- not a polynomial!•but... a polynomial substitution is enough for

UG

(of known UG instances)

(A little) more about our second result2.New (indirect) evidence: natural generalization of UG

•still has ABS-type subexp-time algorithm•cannot be solved in poly-time assuming ETH

Hypercontractive Norm Problem• For any matrix A, q > 2 (even number),

define

• Problem: to approximate

• A natural generalization of the Small Set Expansion (SSE) problem, which is closely related to UG [RS '09]

• Theorem. Constant approximation for 2->q norm is as hard as SSE

qxxqAxA

1:22

max

qA

2

Hypercontractive Norm Problem (cont'd)

• Theorem. 2->q norm can be "well approximated" in time exp(n^(2/q)). Moreover, the algorithm can be used to recover the subexp time algorithm for SSE.– Proof idea: subspace enumeration

• Theorem. Assuming the Exponential Time Hypothesis (i.e. 3-SAT does not have subexp time algorithm), there is no poly-time constant approximation algorithm for 2->q norm.– Proof idea: through quantum information

theory [BCY '10]

Summary

• SoS hierarchy refutes all known UG instances– certain types of soundness proof does not

work for showing a gap of SoS hierarchy

• New connection between hypercontractivity, small set expansion and... quantum separability

Open problems

• Show SoS hierarchy refutes Max-Cut instances?

• More study on 2->q norms...– New UG hard instances from hardness of 2-

>q norms?– Stronger 2->q hardness results (NP-

Hardness)?– dequantumize the current 2->q hardness

proof?

Thank you!