Yuan Zhou Carnegie Mellon University Joint works with Boaz Barak, Fernando G.S.L. Brandão, Aram W. Harrow, Jonathan Kelner, Ryan O'Donnell and David Steurer

Constraint Satisfaction Problems Given: –a set of variables: V –a set of values: Ω –a set of "local constraints": E Goal: find an assignment σ : V -> Ω to maximize #satisfied constraints in E α-approximation algorithm: always outputs a solution of value at least α*OPT

Example 1: Max-Cut Vertex set: V = {1, 2, 3,..., n} Value set: Ω = {0, 1} Typical local constraint: (i, j) э E wants σ(i) ≠ σ(j) Alternative description: –Given G = (V, E), divide V into two parts, –to maximize #edges across the cut Best approx. alg.: approx. [GW'95] Best NP-hardness: [Has'01, TSSW'00]

Example 2: Balanced Seperator Vertex set: V = {1, 2, 3,..., n} Value set: Ω = {0, 1} Minimize #satisfied local constraints: (i, j) э E : σ(i) ≠ σ(j) Global constraint: n/3 ≤ |{i : σ(i) = 0}| ≤ 2n/3 Alternative description: –given G = (V, E) –divide V into two "balanced" parts, –to minimize #edges across the cut

Example 2: Balanced Seperator (cont'd) Vertex set: V = {1, 2, 3,..., n} Value set: Ω = {0, 1} Minimize #satisfied local constraints: (i, j) э E : σ(i) ≠ σ(j) Global constraint: n/3 ≤ |{i : σ(i) = 0}| ≤ 2n/3 Best approx. alg.: sqrt{log n}-approx. [ARV'04] Only (1+ε)-approx. alg. is ruled out even assuming 3-SAT does not have subexp time alg. [AMS'07]

Example 3: Unique Games Vertex set: V = {1, 2, 3,..., n} Value set: Ω = {0, 1, 2,..., q - 1} Maximize #satisfied local constraints: (i, j) э E : σ(i) - σ(j) = c (mod q) Unique Games Conjecture (UGC) [Kho'02, KKMO'07] No poly-time algorithm, given an instance where optimal solution satisfies (1-ε) constraints, finds a solution satisfying ε constraints Stronger than (implies) "no constant approx. alg."

Example 3: Unique Games (cont'd) Vertex set: V = {1, 2, 3,..., n} Value set: Ω = {0, 1, 2,..., q - 1} Maximize #satisfied local constraints: (i, j) э E : σ(i) - σ(j) = c (mod q) UG(ε): to tell whether an instance has a solution satisfying (1-ε) constraints, or no solution satisfying ε constraints Unique Games Conjecture (UGC). UG(ε) is hard for sufficiently large q

Example 3: Unique Games (cont'd) Implications of UGC –For large class of problems, BASIC-SDP (semidefinite programming relaxation) achieves optimal approximation ratio Max-Cut: approx. Vertex-Cover: 2-approx. Max-CSP [KKMO '07, MOO '10, KV '03, Rag '08]

Open questions Is UGC true? Are the implications of UGC true? –Is Max-Cut hard to approximate better than 0.878? –Is Balanced Seperator hard to approximate with in constant factor?

SDP Relaxation hierarchies A systematic way to write tighter and tighter SDP relaxations Examples –Sherali-Adams+SDP [SA'90] –Lasserre hierarchy [Par'00, Las'01] … ? UG(ε) rounds SDP relaxation in roughly time BASIC-SDP GW SDP for Maxcut (0.878-approx.) ARV SDP for Balanced Seperator

How many rounds of tighening suffice? Upperbounds – rounds of SA+SDP suffice for UG(ε) [ABS'10, BRS'11] Lowerbounds [KV'05, DKSV'06, RS'09, BGHMRS '12] (also known as constructing integrality gap instances) – rounds of SA+SDP needed for UG(ε) – rounds of SA+SDP needed for better-than approx for Max-Cut – rounds for SA+SDP needed for constant approx. for Balanced Seperator

Our Results We study the performance of Lasserre SDP hierarchy against known lowerbound instances for SA+SDP hierarchy, and show that 8-round Lasserre solves the Unique Games lowerbound instances [BBHKS Z '12] 4-round Lasserre solves the Balanced Seperator lowerbound instances [O Z '12] Constant-round Lasserre gives better-than approximation for Max-Cut lowerbound instances [O Z '12]

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 ("Sum-of-squares proof system") –lift the soundness proof to the proof system

Remarks Using a connection between SDP hierarchies and algebraic proof systems, we refute all known UG lowerbound instances and many instances for its related problems We provide new insight in designing integrality gap instances -- should avoid soundness proofs that can be lifted to the powerful Sum-of- Squares proof system We show that Lasserre is strictly stronger than other hierarchies on UG and its related problems (as it was believed to be)

Outline of the rest of the talk Sum-of-Squares proof system and Lasserre hierarchy Lift the soundness proofs to the SoS proof system

Sum-of-Squares proof system and Lasserre hierarchy

Polynomial optimization Maximize/Minimize Subject to all functions are low-degree n-variate polynomial functions Max-Cut example: Maximize s.t.

Polynomial optimization (cont'd) Maximize/Minimize Subject to all functions are low-degree n-variate polynomial functions Balanced Seperator example: Minimize s.t.

Certifying no good solution Maximize Subject to To certify that there is no solution better than, simply say that the following equations & inequalities are infeasible

The Sum-of-Squares proof system To show the following equations & inequalities are infeasible, Show that where is a sum of squared polynomials, including 's A degree-d "Sum-of-Squares" refutation, where

Example 1 To refute We simply write A degree-2 SoS refutation

Example 2: Max-Cut on triangle graph To refute We "simply" write......

Example 2: Max-Cut on triangle graph (cont'd) A degree-4 SoS refutation

Relation between SoS proof system and Lasserre SDP hierarchy

Finding SoS refutation by SDP A degree-d SoS refutation corresponds to solution of an SDP with variables The SDP is the same as the dual of -round Lasserre relaxation An SoS refutation => upperbound on the dual of optimum of Lasserre => upperbound on the value of Lasserre –e.g. 4-round Lasserre says that Max-Cut of the triangle graph is at most 2 (BASIC-SDP gives 9/4) Bounding SDP value by SoS refutation

Remarks Positivestellensatz. [Krivine'64, Stengle'73] If the given equalities & inequalities are infeasible, there is always an SoS refutation (degree not bounded). The degree-d SoS proof system was first proposed by Grigoriev and Vorobjov in 1999 Grigoriev showed degree is needed to refute unsatisfiable sparse -linear equations –later rediscovered by Schoenbeck in Lasserre world

SoS proofs (in contrast to refutations) Given assumptions to prove that A degree-d SoS proof writes where are sums of squared polynomials Remark. Degree-d SoS proof => degree-d SoS refutation for

Technical Part: Lift the proofs to SoS proof system

Components of the soundness proof Cauchy-Schwarz/Hölder's inequality Hypercontractivity inequality Smallsets expand in the noisy hypercube Invariance Principle Influence decoding (of known UG instances)

Hypercontractivity Inequality 2->4 hypercontractivity inequality: for low degree polynomial we have Goal of an SoS proof: write Note that 's are indeterminates

Traditional proof of hypercontractivity 2->4 hypercontractivity inequality: for low degree polynomial we have (Traditional) proof. Apply induction on d and n. –Let –g and h are (n-1)-variate polynomials,

Traditional proof of hypercontractivity (cont'd) (Cauchy-Schwartz) (induction) All equalities are polynomial identities about indeterminates

SoS proof of hypercontractivity? The square-root in the Cauchy-Schwartz step looks difficult for polynomials Solution: Prove a stronger statement -- two- function hypercontractivity inequality Theorem. Suppose then

SoS proof of two-fcn hypercontractivity Write using (induction) unroll the induction to get the SoS proof

Components of the soundness proof Cauchy-Schwarz/Hölder's inequality Hypercontractivity inequality Smallsets expand in the noisy hypercube Invariance Principle Influence decoding (of known UG instances)

Smallset expansion of noisy hypercube For, let Theorem. If then Traditional proof. Let be the projection operator onto the eigenspace of with eigenvalue. I.e. the space spanned by

Traditional proof of SSE of noisy hypercube (cont'd) (SoS friendly) (Holder's) (SoS friendly) (hypercontractivity) (SoS friendly) (poly. identity)

Traditional proof of SSE of noisy hypercube (cont'd) (SoS friendly) (take ) Key problem: fractional power involved in the Holder's step Solution: Cauchy-Schwartz/Holders with no fractional power

SoS-izable Cauchy-Schwartz Theorem. For any constant a > 0 where SoS is a sum of squared polynomials of degree at most 2 Remark. and the equality holds when Proof. Skipped. Corollary. (Holder's) For any constant a > 0 Proof. Apply C-S twice

SoS proof of SSE (Holder's) (SoS friendly) (take ) (hypercontractivity)

SoS proof of SSE (cont'd) (take )

Components of the soundness proof Cauchy-Schwarz/Hölder's inequality Hypercontractivity inequality Smallsets expand in the noisy hypercube Invariance Principle Influence decoding (of known UG instances)

A few words on Invariance Principle trickier "bump function" is used in the original proof --- not a polynomial! but... a polynomial substitution is enough for UG

Max-Cut and Balanced Seperator An SoS proof for "Majority Is Stablest" theorem is needed for Max-Cut instances –We don't know how to get around the bump function issue in the invariance step –Instead, we proved a weaker theorem: "2/pi theorem" -- suffices to give better-than algorithms for known Max-Cut instances Balanced Seperator. Key is to SoS-ize the proof for KKL theorem –Hypercontractivity and SSE is also useful there –Some more issues to be handled

Summary SoS/Lasserre hierarchy refutes all known UG instances and Balanced Seperator instances, gives better-than approximation for known Max- Cut instances, –certain types of soundness proof does not work for showing a gap of SoS/Lasserre hierarchy

Open problems Show that SoS/Lasserre hierarchy fully refutes Max-Cut instances? –SoS-ize Majority Is Stablest theorem... More lowerbound instances for SoS/Lasserre hierarchy?

Thank you!