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 & Princeton • David Steurer • MSR New England U NY area Theory Day, November 2010 WARNING: Some constants have been harmed in the making of this talk.

2. Unique Games Input: list of constraints of form Unique Games Conjecture (UGC) [Khot’02] Goal: satisfy as many constraints as possible 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 Implications of UGC: For many basic optimization problems, it is NP-hard to beat current algorithms (based on simple LP or SDP relaxations) Vertex Cover[Khot-Regev’03], Max Cut[Khot-Kindler-Mossel-O’Donnell’04, Mossel-O’Donnell-Oleszkiewicz’05], every Max Csp[Raghavendra’08], …

3. Input:Unique Gamesinstance including constraints of formform with (say) YES:more than of constraints satisfiable Distinguish: NO: less than of constraints satisfiable Our Result: 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:

4. Consequences for unique games conjecture: Subexponential Algorithm for Unique Games NP-hardness reduction for must have blow-up (*)  rules out certain classes of reductions for proving UGC in time 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, exp(no(1))

5. 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 )

6. Talk Outline • Review of graph expansion. • algorithm for : • Decomposition strategy • Graph decomposition lemma • Enumeration algorithm for quasi-expanders. • Conclusions and open problems.

7. Interlude: Graph Expansion -regular graph # edges leaving expansion() = normalized adjacency matrix (stochastic) Eigenvalues of : expansion() Cheeger’s Inequality: [Cheeger70,Alon-Milman84,Alon86] Def: is -expander if Def: is -quasi-expander if

8. Subexponential Algorithm for Unique Games in time General strategy: Partition input to many easy instances, then solve each instance separately. [Trevisan’05, Arora-Impagliazzo-Matthews-Steurer’10] remove 1% of constraints easy easy arbitrary instance easy easy easy easy easy For ¾ of instances For ¾ of instances

9. Subexponential Algorithm for Unique Games in time General strategy: Partition input to many easy instances, then solve each instance separately. [Trevisan’05, Arora-Impagliazzo-Matthews-Steurer.’10] Easy instances of Unique Games: in time if is -expander [Arora-Khot-Kolla-Steurer-Tulsiani-Vishnoi’08] • in time if is -quasi-expander [Kolla-Tulsiani’07, Kolla’10, here] • All results use same “subspace enumeration” algorithm [Kolla-Tulsiani’07] Constraint Graph variable  vertex constraint  edge

10. Subexponential Algorithm for Unique Games in time General strategy: Partition input to many easy instances, then solve each instance separately. [Trevisan’05, Arora-Impagliazzo-Matthews-Steurer.’10] Easy instances of Unique Games: • in time if is -quasi-expander [Kolla-Tulsiani’07, Kolla’10, here] Graph Decomposition Idea: quasi-expander small-set expander Here Classical [Leighton-Rao’88, Goldreich-Ron’98, Spielman-Teng’04, Trevisan’05] By removing 1% of edges, every graph can be decomposed into components with eigenvalue gap at most eigenvalues

11. Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with at most eigenvalues Assume graph is d-regular general For , “higher-order Cheeger bound” [here] follows from Cheeger bound [Cheeger’70, Alon-Milman’85, Alon’86] V eigenvalues eigenvalue gap S smallnon-expanding set non-expanding set expansion() expansion() Continue steps, accumulate fraction of edges Continue steps, accumulate fraction of edges

12. “higher-order Cheeger bound” eigenvalues expansion()

13. “higher-order Cheeger bound” expansion() eigenvalues

14. “higher-order Cheeger bound” expansion() eigenvalues Idea

15. “higher-order Cheeger bound” expansion() eigenvalues It would suffice to show: ( for degree ) vertex such thatfor volume growth in some intermediate step (otherwise ) has expansion

16. “higher-order Cheeger bound” expansion() eigenvalues Suffices collision probability of t-step random walk from i It would suffice to show: vertex such thatfor Heuristic: collision probability collision probability decay in intermediate step s < t level set of has expansion and size

17. “higher-order Cheeger bound” expansion() eigenvalues Suffices collision probability of t-step random walk from i It would suffice to show: vertex such thatfor Heuristic: collision probability “A Markov chain with many large eigenvalues cannot mix locally everywhere” collision probability decay in intermediate step s < t level set of has expansion and size

18. Subexponential Algorithm for Unique Games in time General strategy: Partition input to many easy instances, then solve each instance separately. [Trevisan’05, Arora-Impagliazzo-Matthews-Steurer.’10] Easy instances of Unique Games: • in time if is -quasi-expander [Kolla-Tulsiani’07, Kolla’10, here] Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with at most eigenvalues

19. Assume:label-extended graph has at most eigenvalues constraint graph label-extended graph cloud cloud Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] in time if at most eigenvalues if a-b=c mod k assignment satisfying of constraints vertex set of size and expansion

20. Assume:label-extended graph has at most eigenvalues constraint graph label-extended graph Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] in time if at most eigenvalues assignment satisfying of constraints vertex set of size and expansion enumerate subspace assignment satisfying 90%of constraints 99% of indicator vector lies in span of top m eigenvectors

21. Assume:label-extended graph has at most eigenvalues constraint graph label-extended graph normalized indicator vector Suppose:> 1% of is orthogonal to span Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] in time if at most eigenvalues assignment satisfying of constraints vertex set of size and expansion expansion enumerate subspace assignment satisfying 90%of constraints 99% of indicator vector lies in span of top m eigenvectors Compare: Fourier-based learning

22. Assume:label-extended graph has at most eigenvalues constraint graph label-extended graph Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] in time if at most eigenvalues [Kolla-Tulsiani 07]: If is -expander then is -quasi-expander [Kolla10]: If is -quasi-expander and * then is -quasi-expander Here: If is -quasi-expander then is -quasi-expander Idea: Every cycle in corresponds to cycle in . Implies that

23. More Subexponential Algorithms Similar approximation for Multi CutandSmall Set Expansion Better approximations for Max Cut and Vertex Cover on small-set expanders 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 refuting the Unique Games Conjecture How many large eigenvalues can a small-set expander have? Is Boolean noise graph the worst case? (large eigenvalues) Thank you! Questions?

27. Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with at most eigenvalues Assume graph is d-regular For , follows from Cheeger bound [Cheeger’70, Alon-Milman’85, Alon’86] V eigenvalue gap S non-expanding set expansion() Continue steps, accumulate fraction of edges

28. Graph Decomposition By removing 1% of edges, every graph can be decomposed into components with at most eigenvalues Assume graph is d-regular general For , “higher-order Cheeger bound” [here] follows from Cheeger bound V eigenvalues S smallnon-expanding set expansion() Continue steps, accumulate fraction of edges

29. Subexponential Algorithm for Unique Games in time Easy instances of Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] in time if eigenvalue gap Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] in time if at most eigenvalues quasi-expander Graph Decomposition Idea: quasi-expander small-set expander Here Classical By removing 1% of edges, every graph can be decomposed into components with eigenvalue gap at most eigenvalues