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

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

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

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

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], …

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

Subexponential Algorithm for Unique Games Time vs Approximation Trade-off

Subexponential Algorithm for Unique Games Consequences

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

Subexponential Algorithm for Unique Games here: via semidefinite programming hierarchies

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

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

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

Important fact: (Pairwise) Correlation RoundingConditioningPartitioning

RoundingConditioningPartitioning

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

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

RoundingConditioningPartitioning Issue: Wanted:

RoundingConditioningPartitioning Correlation Propagation

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]

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]