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Yuan Zhou Carnegie Mellon University

This work explores the approximability of constraint satisfaction problems and the complexity of proving the hardness of approximation. It focuses on problems such as Max-Cut, Balanced Separator, and Unique Games. The study involves the performance of the Lasserre SDP hierarchy against known lower bound instances, providing new insights into integrality gap instances.

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Yuan Zhou Carnegie Mellon University

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  1. Approximability and Proof Complexity 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

  2. 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

  3. 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.: 0.878-approx.[GW'95] • Best NP-hardness: 0.941[Has'01, TSSW'00]

  4. 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

  5. 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]

  6. 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."

  7. 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

  8. 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: 0.878-approx. Vertex-Cover: 2-approx. Max-CSP [KKMO '07, MOO '10, KV '03, Rag '08]

  9. 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?

  10. 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] BASIC-SDP rounds SDP relaxation in roughly time ? … ARV SDP for Balanced Seperator UG(ε) GW SDP for Maxcut (0.878-approx.)

  11. 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-0.878 approx for Max-Cut • rounds for SA+SDP needed for constant approx. for Balanced Seperator

  12. 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 [BBHKSZ'12] • 4-round Lasserre solves the Balanced Seperator lowerbound instances [OZ'12] • Constant-round Lasserre gives better-than-0.878 approximation for Max-Cut lowerbound instances [OZ'12]

  13. 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

  14. 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

  15. 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)

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

  17. Sum-of-Squares proof system and Lasserre hierarchy

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

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

  20. 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

  21. 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

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

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

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

  25. Relation between SoS proof system and Lasserre SDP hierarchy

  26. 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

  27. 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

  28. 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

  29. Technical Part:Lift the proofs to SoS proof system

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

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

  32. 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,

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

  34. 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

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

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

  37. 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

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

  39. 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

  40. 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

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

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

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

  44. 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

  45. 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-0.878 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

  46. Summary • SoS/Lasserre hierarchy refutes all known UG instances and Balanced Seperator instances, gives better-than-0.878 approximation for known Max-Cut instances, • certain types of soundness proof does not work for showing a gap of SoS/Lasserre hierarchy

  47. 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?

  48. Thank you!

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