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
xi  y j  c (mod q)
– Distinguish
• YES case: at least (1-ε) of the equations
are satisfiable
• NO case: at most ε of the equations are
satisfiable
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]
2.New (indirect) evidence:
natural generalization of UG
•still has ABS-type subexp-time algorithm
•cannot be solved in poly-time assuming ETH
• Evidences opposing UGC
– Subexponential time algorithms solve UG [ABS '10]
1.New evidence: a natural SDP hierarchy
solves all previous "hard instances" for UG
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 SDPO (rrelaxation
)
in roughly n
time
?
…
e.g. [SA'90,LS'91]
UG(ε)
• For certain SDP relaxation hierarchies
  (1)
– Upper bound : n
rounds suffice
[ABS'10,BRS'11,GS'11]
– Lower bound : exp((loglog n)(1) ) rounds needed
[RS'09,BGHMRS'11]
Semidefinite programming (SDP) relaxation hierarchies
(cont'd)
r rounds SDPO (rrelaxation
)
in roughly n
time
?
…
e.g. [SA'90,LS'91]
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
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
Axioms
p1 ( x)  0
p2 ( x )  0

pm ( x )  0
Goal
derive
q ( x)  0
"Positivstellensatz"
[Stengle'74]
• Rules
– p1 ( x)  0, p2 ( x)  0  p1 ( x)  p2 ( x)  0
– p1 ( x)  0, p2 ( x)  0  p1 ( x) p2 ( x)  0
– h( x)2  0
– all intermediate polynomials have bounded
degree d ("level-d" SoS proof system)
Example
Axioms
xx 0
Goal
1 x  0
derive
2
• SoS proof:
1  x  ( x  x )  (1  x)  0
2
2
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
(of known UG instances)
• "Easy" parts
– Cauchy-Schwarz/Hölder's inequality
– Influence decoding
• "Hard" parts
– Hypercontractivity inequality
– Invariance Principle
SoS proof of hypercontractivity
• The 2->4 hypercontractivity inequality:
 S  xi
for low degree polynomial f ( x) 

S [ n ],|S |d
1/ 4
we have 
E n[ f ( x) 4 ] 
 x{1,1}

iS
1/ 2
 3  E n[ f ( x) 2 ] 
 x{1,1}

d
• The goal of an SoS proof is to show
2
9d  E n[ f ( x) 2 ]   E n[ f ( x) 4 ]
is sum of squares
 x{1,1}
 x{1,1}
• Prove by induction (very similar to the wellknown inductive proof of the inequality itself)...
Components of the soundness proof
(of known UG instances)
• "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
(A little) more about our second result
2.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
A 2q  max Ax q
x: x 2 1
• Problem: to approximate A
2 q
• 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
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 (NPHardness)?
– dequantumize the current 2->q hardness
proof?
Thank you!