Boaz Barak (MSR New England)Fernando G.S.L. Brando (Universidade Federal de Minas Gerais)Aram W. Harrow (University of Washington)Jonathan Kelner (MIT)David Steurer (MSR New England)Yuan Zhou (CMU)

MotivationUnique Games Conjecture (UGC) [Khot'02]For every constant , the following problem is NP-Hard

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

DistinguishYES case: at least (1-) of the equations are satisfiableNO case: at most of the equations are satisfiable

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

Implications of UGCFor 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 resultsEvidences in favor of UGC(Certain) strong SDP relaxation hierarchies cannot solve UG in polynomial time [RS'09, KPS'10, BGHMRS'11]

Evidences opposing UGCSubexponential time algorithms solve UG [ABS '10]

Our resultsEvidences in favor of UGC(Certain) strong SDP relaxation hierarchies cannot solve UG in polynomial time [RS'09, KPS'10, BGHMRS'11]

Evidences opposing UGCSubexponential time algorithms solve UG [ABS '10] 1.New evidence: a natural SDP hierarchy solves all previous "hard instances" for UG2.New (indirect) evidence: natural generalization of UGstill has ABS-type subexp-time algorithmcannot be solved in poly-time assuming ETH

More about our first result1.New evidence: a natural SDP hierarchy solves all previous "hard instances" for UG

Semidefinite programming (SDP) relaxation hierarchies? rounds SDP relaxation in roughly timee.g. [SA'90,LS'91]UG()For certain SDP relaxation hierarchiesUpper bound : rounds suffice [ABS'10,BRS'11,GS'11]Lower bound : rounds needed [RS'09,BGHMRS'11]

Semidefinite programming (SDP) relaxation hierarchies?UG()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) rounds SDP relaxation in roughly timee.g. [SA'90,LS'91]

Proof overview Integrality gap instanceSDP completeness: a good vector solutionIntegral soundness: no good integral solution

A common method to construct gaps (e.g. [RS'09])Use the instance derived from a hardness reductionLift the completeness proof to vector worldUse the soundness proof directly

Proof overview (cont'd)Our goal: to prove there is no good vector solutionRounding algorithms?

Instead, we bound the value of the dual of the SDPinterpret the dual of the SDP as a proof systemlift the soundness proof to the proof system

Sum-of-Squares proof systemAxiomsGoalderiveRules all intermediate polynomials have bounded degree d ("level-d" SoS proof system) "Positivstellensatz" [Stengle'74]

ExampleSoS proof:AxiomsGoalderiveAxiomsquared termFormulate 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" partsCauchy-Schwarz/Hlder's inequalityInfluence decoding

"Hard" partsHypercontractivity inequalityInvariance Principle(of known UG instances)

SoS proof of hypercontractivityThe 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)...

Components of the soundness proof"Easy" partsCauchy-Schwarz/Hlder's inequalityInfluence decodingIndependent rounding

"Hard" partsHypercontractivity inequalityInvariance Principletrickier "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 UGstill has ABS-type subexp-time algorithmcannot be solved in poly-time assuming ETH

Hypercontractive Norm ProblemFor 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

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]

SummarySoS hierarchy refutes all known UG instancescertain 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 problemsShow 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!