Boaz Barak ( MSR New England )
description
Transcript of Boaz Barak ( MSR New England )
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!