1. The Unique Games Conjecture and Graph Expansion School on Approximability, Bangalore, January 2011 Joint work with S Prasad Raghavendra Georgia Institute of Technology David Steurer Microsoft Research New England

2. The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

3. The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

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

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

6. 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], …

7. What are hard instances for U NIQUE G AMES ?

8. Random instances Random 3-S AT Hypothesis [Appelbaum-Barak-Wigderson’10, Bhaskara- -Charikar-Chlamtac-Feige-Vijayaraghavan’10] Other problems: Planted D ENSEST k-S UBGRAPH U NIQUE G AMES :  expanding constraint graph [Arora-Khot-Kolla-S.-Tulsiani-Vishnoi’08]  few large eigenvalues [Kolla’10, Barak-Raghavendra-S.’11]  strong small-set expanders [Arora-Impagliazzo-Matthews-S.’10] [Feige’02, Schoenebeck’08] Easy for random-like instances

9. What are hard instances for U NIQUE G AMES ? Random 3-S AT,Planted D ENSEST k-S UBGRAPH U NIQUE G AMES :Easy for random-like instances Other problems: Random instances Other problems: C LIQUE on product graphs P ROJECTION GAMES from parallel repetition [Raz’98] U NIQUE G AMES : Easy for parallel-repeated instances of M AX C UT [Barak-Hardt-Haviv-Rao-Regev-S.’08] (based on counterexample for strong parallel repetition [Raz’08]) Combinations of mildly hard instances

10. C LIQUE, U NIQUE G AMES : P ROJECTION GAMES Easy for parallel-repeated instances of M AX C UT What are hard instances for U NIQUE G AMES ? Combinations of mildly hard instances Random 3-S AT,Planted D ENSEST k-S UBGRAPH U NIQUE G AMES :Easy for random-like instances Other problems: Random instances Other problems: Hard instances for U NIQUE G AMES from hard instances for S MALL S ET E XPANSION [here] natural generalization of S PARSEST C UT

11. The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

12. d-regular graph G d vertex set S Graph Expansion expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| Important notion in many contexts: derandomization, network routing, coding theory, Markov chains, differential geometry, group theory

13. S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V|

14. S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| S PARSEST C UT Goal:find S with volume(S) < ½ so as to minimize expansion(S) Input:graph G eigenvalue gap: first non-trivial approximation [Cheeger’70] No strong hardness for approximating S PARSEST C UT known! (even assuming UGC) Approximating S PARSEST C UT

15. S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| S PARSEST C UT Goal:find S with volume(S) < ½ so as to minimize expansion(S) Input:graph G S MALL -S ET E XPANSION no poly-time algorithm with non-trivial approximation guarantee known Goal: Distinguish two cases Small Set Expansion Hypothesis (SSE)

16. S expansion(S) = # edges leaving S d |S| volume(S ) = |S| |V| S PARSEST C UT Goal:find S with volume(S) < ½ so as to minimize expansion(S) Input:graph G S MALL -S ET E XPANSION no poly-time algorithm with non-trivial approximation guarantee known Goal: Distinguish two cases Small Set Expansion Hypothesis (SSE)

17. The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

18. “Superficial” Connection of U NIQUE G AMES and S MALL -S ET E XPANSION

19. SSE UGC + SSE [Raghavendra-S.’10] UGC additional promise: small-set expansion of constraint graph M AX C UT V ERTEX C OVER Any M AX C SP … S PARSEST C UT B ALANCED S EPARATOR M INIMUM L INEAR A RRANGEMENT M IN k-C UT … [Raghavendra- -Tulsiani-S.’10] [Raghavendra- -Tulsiani-S.’10]

20. graph G A B Verifier BA Prover 1Prover 2 ba Verifier To show: a b

21. graph G A B

22. S A B Completeness Partial Strategy for Prover 1 (and 2) : (otherwise, refuse to answer)

23. graph G Soundness S

24. The Unique Games Conjecture Graph Expansion Reductions Integrality Gaps

25. Goal: rule out that certain (classes of) algorithms refute the UGC or the SSE hypothesis Here:algorithms based on (hierarchies) of SDP relaxations (capture current best approximation algorithms)

26. Basic SDP relaxation for S MALL -S ET E XPANSION minimize subject to

27. Basic SDP relaxation for S MALL -S ET E XPANSION minimize subject to Integrality Gap Instance

31. Integrality Gaps for U NIQUE G AMES via reductions  super-polynomial lower-bound for U NIQUE G AMES in a restricted computational model

32. Integrality Gap Instance Integrality Gaps for U NIQUE G AMES via reductions Thank you!Questions?