1. Subexponential Algorithms for Unique Games and Related Problems Approximation Algorithms, June 2011 David Steurer MSR New England Sanjeev Arora Princeton University & CCI Boaz Barak MSR New England U

2. Subexponential Algorithms for Unique Games and Related Problems David Steurer MSR New England Sanjeev Arora Princeton University & CCI Boaz Barak MSR New England Rounding Semidefinite Programming Hierarchies via Global Correlation David Steurer MSR New England Prasad Raghavendra Georgia Tech Boaz Barak MSR New England

3. U NIQUE G AMES Goal:satisfy as many constraints as possible

4. U NIQUE G AMES Goal:satisfy as many constraints as possible Goal: Distinguish two cases Unique Games Conjecture (UGC) [Khot’02]

5. Implications of UGC For many basic optimization problems, it is NP-hard to beat current algorithms (based on simple LP or SDP relaxations) Examples: V ERTEX C OVER [Khot-Regev’03], M AX C UT [Khot-Kindler-Mossel-O’Donnell’04, Mossel-O’Donnell-Oleszkiewicz’05], every M AX C SP [Raghavendra’08], …

6. Implications of UGC For many basic optimization problems, it is NP-hard to beat current algorithms (based on simple LP or SDP relaxations) Unique Games Barrier U NIQUE G AMES is common barrier for improving current algorithms of many basic problems

7. Subexponential Algorithm for Unique Games Time vs Approximation Trade-off

8. Subexponential Algorithm for Unique Games Consequences

9. 2-S AT 3-S AT (*) F ACTORING [Moshkovitz-Raz’08 + Håstad’97] Subexponential Algorithm for Unique Games G RAPH I SOMORPHISM

10. Subexponential Algorithm for Unique Games here: via semidefinite programming hierarchies

11. What we want: U NIQUE G AMES Goal:satisfy as many constraints as possible

12. Goal: Here: [Arora-Barak-S.’10 + Barak-Raghavendra-S.’11] consistency positive semidefiniteness What we want:

13. Components of iterative procedure Rounding Conditioning sample variables independently according to their marginals Partitioning enough if constraint graph has few significant eigenvalues [BRS’11] general framework for rounding SDP hierarchies

14. Important fact: (Pairwise) Correlation RoundingConditioningPartitioning

15. RoundingConditioningPartitioning

16. sample variables independently according to their marginals Local Correlation (over edges of constraint graph) RoundingConditioningPartitioning

17. computationally expensive Global Correlation (over random vertex pairs) RoundingConditioningPartitioning Issue: Idea:

18. RoundingConditioningPartitioning Issue: Wanted:

19. RoundingConditioningPartitioning Correlation Propagation

20. For general 2-C SP : QPTAS if constraint graph is hypercontractive very good expander for small sets [Barak-Raghavendra-S.’11] Subsequent work: [Arora-Ge’11] better 3-C OLORING approximation on some graph families Independent work: approximation schemes for quadratic integer programming with p.s.d. objective & few relevant eigenvalues More SDP-hierarchy algorithms [Guruswami-Sinop’11]

21. Open Questions How many large eigenvalues can a small-set expander have? Thank you!Questions? What else can be done in subexponential time? Towards settling the Unique Games Conjecture Better approximations for M AX C UT or V ERTEX C OVER on general instances? [Barak-Gopalan-Håstad-Meka-Raghavendra-S.’11]