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 School on Approximability, Bangalore, January 2011

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

3. 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

4. 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], …

5. 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

6. 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:

7. 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, exp(no(1))

8. 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 )

9. Subexponential Algorithm for Unique Games in time Interlude: Graph Expansion -regular graph # edges leaving expansion() = normalized adjacency matrix (stochastic) normalized indicator vector expansion()

10. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] in time if eigenvalue gap Constraint Graph variable  vertex constraint  edge

11. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] in time if eigenvalue gap [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues in time if at most eigenvalues quasi-expander Graph Eigenvalues normalized adjacency matrix …

12. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] in time if eigenvalue gap [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues in time if at most eigenvalues quasi-expander Subspace Enumeration [Kolla-Tulsiani’07] exhaustive searchovercertain subspace Question: Dimension of this subspace?

13. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] easy remove 1% of edges easy in time if eigenvalue gap hard constraint graph easy easy easy [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues easy easy in time if at most eigenvalues quasi-expander Graph Decomposition Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] By removing 1% of edges, decompose constraint graph into components for which is “easy”

14. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] opt>1- opt>1- in time if eigenvalue gap YES: opt>1- opt>1- Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] opt>1- in time if at most eigenvalues quasi-expander Graph Decomposition :Given UG instance, distinguish YES: NO: Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] By removing 1% of edges, decompose constraint graph into components for which is “easy”

15. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] YES YES in time if eigenvalue gap YES: YES ? YES Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] ? YES in time if at most eigenvalues Graph Decomposition :Given UG instance, distinguish YES: NO: Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] By removing 1% of edges, decompose constraint graph into components for which is “easy”

16. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] opt< opt< in time if eigenvalue gap NO: opt< opt< Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] opt< in time if at most eigenvalues Graph Decomposition :Given UG instance, distinguish YES: NO: Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] By removing 1% of edges, decompose constraint graph into components for which is “easy”

17. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] NO NO in time if eigenvalue gap NO: NO ? NO Constraint graph with few large eigenvalues [Kolla-Tulsiani’07, Kolla’10, here] NO ? in time if at most eigenvalues Graph Decomposition :Given UG instance, distinguish YES: NO: Approach [Trevisan’05, Arora-Impagliazzo-Matthews-S.’10] By removing 1% of edges, decompose constraint graph into components for which is “easy”

18. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] in time if eigenvalue gap [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues in time if at most eigenvalues quasi-expander 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

19. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games Expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08] in time if eigenvalue gap [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues 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

20. Assume:label-extended graph has at most eigenvalues 2 constraint graph label-extended graph cloud cloud [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues in time if at most eigenvalues if a-b=c mod k assignment satisfying of constraints vertex set of size and expansion

21. Assume:label-extended graph has at most eigenvalues 2 constraint graph label-extended graph [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues 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

22. Assume:label-extended graph has at most eigenvalues 2 constraint graph label-extended graph normalized indicator vector Suppose:> 1% of is orthogonal to span [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues 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

23. [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues in time if at most eigenvalues

24. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues in time if at most eigenvalues 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

25. Subexponential Algorithm for Unique Games in time Easy graphs for Unique Games [Kolla-Tulsiani’07, Kolla’10, here, Barak-Raghavendra-S.’11] Constraint graph with few large eigenvalues in time if at most eigenvalues 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

26. 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()

27. 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()

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

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

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

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

32. “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 local Cheeger bound collision probability decay in intermediate step s < t level set of has expansion and size

33. “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

34. More Subexponential Algorithms Similar approximation for Multi CutandSmall Set Expansion Better approximations for Max Cut and Vertex Cover on small-set expanders Improved approximations for d-to-1 Games( Khot’sd-to-1 Conjecture) [S.’10] 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?