Yuan Zhou, Ryan O’Donnell Carnegie Mellon University

1. Approximability and Proof Complexity Yuan Zhou, Ryan O’Donnell Carnegie Mellon University

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

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), 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."

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

8. 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 -round SDP relaxation in roughly time ? … ARV SDP for Balanced Separator UG(ε) GW SDP for Maxcut (0.878-approx.)

9. 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-0.878 approx for Max-Cut • rounds for SA+SDP needed for constant approx. for Balanced Separator

10. 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 Gameslowerbound instances

11. 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-0.878 approximation for Max-Cutlowerbound instances • 4-round Lasserre gives constant approximation for the the Balanced Separatorlowerbound instances

12. 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 (”SOSd/sum-of-squares proof system") • lift the soundness proof to the proof system

13. What is the SOSdproof system?

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

15. Polynomial optimization (cont'd) • Maximize/Minimize • Subject to all functions are low-degree n-variatepolynomials • Balanced Separator example: Minimize s.t.

16. 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

17. 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

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

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

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

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

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

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

24. 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-izeRaghavendra’s theorem.

25. Thank you!