Yuan Zhou, Ryan O’Donnell Carnegie Mellon University

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) in 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 Separator Vertex set: V = {1, 2, 3,..., n} Value set: Ω = {0, 1} Minimize #satisfied local constraints: (i, j) in 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 Saperator (cont'd) Vertex set: V = {1, 2, 3,..., n} Value set: Ω = {0, 1} Minimize #satisfied local constraints: (i, j) in 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 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), c} in 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."

Open questions Is UGC true? Is Max-Cut hard to approximate better than 0.878? Is Balanced Separator hard to approximate with in constant factor? Easier questions Do the known powerful optimization algorithms solve UG/Max-Cut/Balanced Separator?

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) -round SDP relaxation in roughly time BASIC-SDP GW SDP for Maxcut (0.878-approx.) ARV SDP for Balanced Separator

How many rounds of tighening suffice? Upperbounds – rounds of SA+SDP suffice for UG [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 – rounds of SA+SDP needed for better-than approx for Max-Cut – rounds for SA+SDP needed for constant approx. for Balanced Separator

From SA+SDP to Lasserre SDP Are the integrality gap instances for SA+SDP also hard for Lasserre SDP? Previous result [BBHKSZ'12] –No for UG –8-round Lasserre solves the Unique Games lowerbound instances

From SA+SDP to Lasserre SDP (cont’d) Are the integrality gap instances for SA+SDP also hard for Lasserre SDP? This paper –No for Max-Cut and Balanced Separator –Constant-round Lasserre gives better-than approximation for Max-Cut lowerbound instances –4-round Lasserre gives constant approximation for the the Balanced Separator lowerbound instances

Proof overview Integrality gap instance –SDP completeness: good vector solution –Integral soundness: no good integral solution Show the instance is not integrality gap instance for Lasserre SDP – no good vector solution –we bound the value of the dual of the SDP –interpret the dual as a proof system (”SOS d /sum-of-squares proof system") –lift the soundness proof to the proof system

What is the SOS d proof system?

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

Polynomial optimization (cont'd) Maximize/Minimize Subject to all functions are low-degree n-variate polynomials Balanced Separator 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 equalities & inequalities are infeasible

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

Positivstellensatz Subject to some mild technical conditions, every infeasible system has such a refutation Caveat: f i ’s and h might need to have high degree. Lasserre SDP and SOS d proof system A degree-d SOS refutation  O(d)-round Lasserre SDP is infeasible

Summary Defined the degree-d SOS proof system Remaining task Integral soundness proof  low-degree refutation in the SOS proof system

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

One-slide How-to Thm: Min-Balanced-Separator in this graph is ≥ blah Proof: … hypercontractivity… “Check out these polynomials.” Thm: Max-Cut of this graph is ≤ blah Proof: … Invariance Principle … … Majority-Is-Stablest… “Check out these polynomials.”

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

Latest results Our theorem on Max-Cut is improved by [DMN’12] –Constant-round Lasserre SDP almost exactly solves the known instances Constant-round Lasserre SDP solves the hard instances for Vertex-Cover [KOTZ’12] Open question Does constant-round Lasserre SDP solve the known instances for all the CSPs? –I.e. SOS-ize Raghavendra’s theorem.

Thank you!