Boaz Barak MSR New England Fernando G S

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

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

Semidefinite programming (SDP) relaxation hierarchies (cont'd) ? … rounds SDP relaxation in roughly time e. g. [SA'90, LS'91] UG(ε) • Theorem. – Sum-of-Squares (So. S) 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 Goal derive "Positivstellensatz" [Stengle'74] • Rules – – all intermediate polynomials have bounded degree d ("level-d" So. S proof system)

Example Axioms derive Goal • So. S proof: Axiom squared term Formulate UG as a polynomial optimization problem, and re-write the existing soundness proof in a So. S 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

So. S proof of hypercontractivity • The 2 ->4 hypercontractivity inequality: for low degree polynomial we have • The goal of an So. S proof is to show is sum of squares • 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 • 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]

Summary • So. S hierarchy refutes all known UG instances – certain types of soundness proof does not work for showing a gap of So. S hierarchy • New connection between hypercontractivity, small set expansion and. . . quantum separability

Open problems • Show So. S 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!