Subexponential Algorithms for Unique Games and Related Problems

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

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

3. Unique Games Input: list of constraints of form Goal: satisfy as many constraints as possible

4. Unique Games Input: list of constraints of form Goal: satisfy as many constraints as possible Unique Games Conjecture (UGC) [Khot’02] For every , the following is NP-hard: Input:Unique Gamesinstance with (say) Goal: Distinguish two cases YES:more than of constraints satisfiable NO: less than of constraints satisfiable

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: Vertex Cover[Khot-Regev’03], Max Cut[Khot-Kindler-Mossel-O’Donnell’04, Mossel-O’Donnell-Oleszkiewicz’05], every Max Csp[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 Example:-approximation for Max Cut at least as hard as Goemans–Williamson bound for Max Cut Unique Games is common barrier for improving current algorithms of many basic problems

7. Subexponential Algorithm for Unique Games in time Time vs Approximation Trade-off Input: Unique Gamesinstance with alphabet size k such that of constraints are satisfiable, Output: assignment satisfying of constraints Time:

8. Subexponential Algorithm for Unique Games Consequences in time NP-hardness reduction for must have blow-up (*)  rules out certain classes of reductions for proving UGC Analog of UGC with subconstant (say ) is false (*) (contrast: subconstant hardness for Label Cover[Moshkovitz-Raz’08]) UGC-based hardness does not rule out subexponential algorithms,  Possibility:-time algorithm for Max Cut() ? (*) assuming 3-Sat does not have subexponential algorithms,

9. Subexponential Algorithm for Unique Games [Moshkovitz-Raz’08 + Håstad’97] Max Cut()? in time Max 3-Sat() Max 3-Sat() Factoring Max Cut() Label Cover() Graph Isomorphism 2-Sat 3-Sat (*) (*) assuming Exponential Time Hypothesis[Impagliazzo-Paturi-Zane’01] (3-Sat has no algorithm )

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

11. Unique Games Input: list of constraints of form Goal: satisfy as many constraints as possible What we want: jointly distributed random variables over for typical constraint

12. distributions over for all with consistency same marginal for in distributions and positive semidefiniteness = indicator of time p.s.d. for all with - -level SDP hierarchy: What we want: -local jointly distributed random variables over for typical constraint Goal: produce global random variables for most constraints iterative procedure for Here: [Arora-Barak-S.’10 + Barak-Raghavendra-S.’11]

13. Components of iterative procedure Rounding sample variables independently according to their marginals Conditioning enough if constraint graph has few significant eigenvalues [BRS’11] pick a vertex and sample condition on sample for Partitioning general framework for rounding SDP hierarchies find vertex subset break dependence between and

14. Rounding Conditioning Partitioning statistical distance between and (Pairwise) Correlation decrease in variance when conditioning on similar to mutual information Important fact: can approximate by Gram matrix of unit vectors (tensoring trick)

15. Rounding Conditioning Partitioning

16. Rounding Conditioning Partitioning sample variables independently according to their marginals statistical distance between independent and correlated sampling If then independent sampling satisfies constraint with probability Rounding fails Local Correlation (over edges of constraint graph)

17. Rounding Conditioning Partitioning pick a vertex and sample condition on sample for Issue: computationally expensive (level level ) measures decrease in variance when conditioning on Idea: condition on vertex only if can condition times on such vertices Conditioning fails Global Correlation (over random vertex pairs)

18. Rounding Conditioning Partitioning find vertex subset break dependence between and Issue: destroy correlation for constraints between and Wanted: set with small expansion & small cardinality

19. Rounding Conditioning Partitioning fails only if local correlation high fails only if global correlation low find vertex subset break dependence between and Correlation Propagation For endpoints of random path of length is Gram matrix of unit vectors random walk stuck for steps on fraction of vertices vertex set with and expansion A vertex is cut in at most partitioning steps break onlyedges

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

21. Open Questions What else can be done in subexponential time? Better approximations for Max Cutor Vertex Cover on general instances? Example:-approximation for Sparsest Cut in time ? Towards settling the Unique Games Conjecture How many large eigenvalues can a small-set expander have? Is Boolean noise graph the worst case? (large eigenvalues) No: small set expander with large eigenvalues [Barak-Gopalan-Håstad-Meka-Raghavendra-S.’11] Thank you! Questions?