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Subexponential Algorithms for Unique Games and Related Problems School on Approximability, Bangalore, January 2011 David Steurer MSR New England Sanjeev Arora Princeton University & CCI Boaz Barak MSR New England & Princeton U

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

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

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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], …

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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

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Subexponential Algorithm for Unique Games Time vs Approximation Trade-off

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(*) assuming 3-S AT does not have subexponential algorithms, exp( n o(1) ) Subexponential Algorithm for Unique Games Consequences

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

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Subexponential Algorithm for Unique Games Interlude: Graph Expansion

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Subexponential Algorithm for Unique Games Easy graphs for U NIQUE G AMES Constraint Graph variable vertex constraint edge Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]

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Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] Subexponential Algorithm for Unique Games Graph Eigenvalues normalized adjacency matrix quasi-expander [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]

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Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Subspace Enumeration [Kolla-Tulsiani’07] exhaustive search over certain subspace Subexponential Algorithm for Unique Games Question: Dimension of this subspace? quasi-expander

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[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] quasi-expander Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] Subexponential Algorithm for Unique Games Graph Decomposition Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] hard constraint graph remove 1% of edges easy

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Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] quasi-expander Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] [Kolla-Tulsiani’07, Kolla’10, here] Subexponential Algorithm for Unique Games Graph Decomposition

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Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] [Kolla-Tulsiani’07, Kolla’10, here] Subexponential Algorithm for Unique Games Graph Decomposition ? YES ?

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Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] [Kolla-Tulsiani’07, Kolla’10, here] Subexponential Algorithm for Unique Games Graph Decomposition

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Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] [Kolla-Tulsiani’07, Kolla’10, here] Subexponential Algorithm for Unique Games Graph Decomposition NO ? ?

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[Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] quasi-expander Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] Subexponential Algorithm for Unique Games Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with Classical Here [Leighton-Rao’88, Goldreich-Ron’98, Spielman-Teng’04, Trevisan’05]

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Expanding constraint graph Constraint graph with few large eigenvalues Easy graphs for U NIQUE G AMES [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] Subexponential Algorithm for Unique Games Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with Classical Here quasi-expander [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]

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Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] 2

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Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] 99% of indicator vector lies in span of top m eigenvectors enumerate subspace 2

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Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] 99% of indicator vector lies in span of top m eigenvectors enumerate subspace Compare: Fourier-based learning 2

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Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11]

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Easy graphs for U NIQUE G AMES Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with Classical Here Subexponential Algorithm for Unique Games

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Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Easy graphs for U NIQUE G AMES Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with Classical Here Subexponential Algorithm for Unique Games

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Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with follows from Cheeger bound [Cheeger’70, Alon-Milman’85, Alon’86] V S Assume graph is d-regular

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Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with Assume graph is d-regular follows from Cheeger bound V S “higher-order Cheeger bound” [here]

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“higher-order Cheeger bound”

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Idea

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“higher-order Cheeger bound” It would suffice to show:

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“higher-order Cheeger bound” It would suffice to show: collision probability of t-step random walk from i Suffices local Cheeger bound

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“higher-order Cheeger bound” It would suffice to show: “A Markov chain with many large eigenvalues cannot mix locally everywhere” collision probability of t-step random walk from i Suffices

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Improved approximations for d- TO -1 G AMES ( Khot’s d-to-1 Conjecture) [S.’10] Better approximations for M AX C UT and V ERTEX C OVER on small-set expanders Open Questions How many large eigenvalues can a small-set expander have? Thank you!Questions? More Subexponential Algorithms Similar approximation for M ULTI C UT and S MALL S ET E XPANSION What else can be done in subexponential time? Towards refuting the Unique Games Conjecture Better approximations for M AX C UT or V ERTEX C OVER on general instances?

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