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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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Subexponential Algorithms for Unique Games and Related Problems Approximation Algorithms, June 2011 David Steurer MSR New England Sanjeev Arora Princeton.

Subexponential Algorithms for Unique Games and Related Problems Approximation Algorithms, June 2011 David Steurer MSR New England Sanjeev Arora Princeton.

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