Near-Optimal Algorithms for Unique Games

1. Near-Optimal Algorithms for Unique Games Yury Makarychev Moses Charikar Konstantin Makarychev Princeton University

2. 8 ( ) d 2 1 3 3 1 7 + ´ x x m o 1 4 > > > ( ) d 1 6 4 1 7 + < ´ x x m o 3 2 > : : : > > ( ) d : 5 3 9 1 7 + ´ x x m o 1 9 Example • Linear equations mod p, two var’s per equation. Maximize # of satisfied equations.

3. Unique Games

4. Unique Games Permutations

5. Unique Games

6. Unique Games

7. Unique Games

8. Unique Games

9. Unique Games Conjecture • Unique Games Conjecture [Khot’02] • instance where 1-fraction of constraints is satisfiable • NP-hard to satisfy even fraction • for every positive constants  and , 9 sufficiently large k. • Used to prove many hardness results

10. Hardness Results Assuming UGC • Max Cut • Vertex Cover • Min 2 CNF Deletion • Multicut • Sparsest Cut • Coloring 3-colorable graphs Match algorithmic results

11. p = 2 1 5 ( ( = ) ) k l O 1 1 ¡ " o g " p ( ) 3 l O 1 ¡ " o g n ; p ( ) [ ] l O G T 1 ( ) ¡ l O 1 ¡ " o g n " o g n Approximation Algorithms Assume 1- fraction is satisfiable. • Random Assignment: 1/k. • Andersson, Engebretsen, Hastad ’01, slightly better than random for lin. eq. • Khot ’02, SDP based algorithm, • Trevisan ’05, SDP based alogrithm, • Gupta and Talwar ’05, LP based,

12. = l k 1 1 0 c o g = 2 1 5 ( ) p k O 1 = ( ) ¡ 2 1 ¡ ¡ ¡ ( ) l k O " 1 " " k k ¡ " o g Our results • Algorithms cover the entire range of . Our Prior

13. = = = l l k k 1 1 1 o o g g n » = = l k 1 1 1 0 ¡ = o g n k l k » o g Comparison For what  does the algorithm satisfy a constant fraction of constraints? Assume k ~ log n.

14. " ¡ p ( ) k l k O 1 ¡ ¡ 2 o r " o g " = = ( ) 4 9 ¡ ¡ + " " o " k k Near Optimality • Khot, Kindler, Mossel and O'Donnell showed that even a slight improvement of our results refutes the UGC. • Khot and Vishnoi constructed an integrality gap instance:

15. Roadmap Integer Program Semidefinite Relaxation Uniform Case Non-uniform Case Conclusions

16. u v u u u v v v 1 2 3 1 2 3 Integer Program

17. u v u u u u v v v v = ( ) i 3 1 2 1 3 2 i ¼ u v Integer Program 0 1 1 0 0 0 we want

18. 1 X u v 2 j j 0 ¡ u v = ( ) i i ¼ 2 u v u u u v v v i 3 2 1 2 3 1 Integer Program 0 1 0 0 1 0 0 0 0

19. 1 X u v 2 j j 1 ¡ u v = ( ) i i ¼ 2 u v u u u v v v i 3 2 1 2 3 1 Integer Program 0 1 0 1 0 0 1 0 1

20. Ã ! k 1 X X 2 j j i ¡ m n u v ( ) i [ ] f g i 8 8 k V i 0 1 ¼ 2 2 2 u u 2 u v i ; i ( ) 1 E 2 = u v ; 8 8 6 V i j 0 2 ¢ u u u = = i j k X 2 8 V 1 2 u u = i i 1 = Integer Program

21. Ã ! k [ ] h i 8 8 k 6 V 1 i j i j 0 2 2 u u u = = X X i j 2 ; ; ; j j i ¡ m n u v ( ) i i ¼ k 2 u v X i ( ) 1 2 E j j 2 8 V = 1 u v 2 u u ; = i i 1 = Semidefinite Program

22. [ [ ] ] P P i i j r x r x x » » = = = u u u ; Ã ! k [ ] h i 8 8 k 6 V 1 i j i j 0 2 2 u u u = = X X i j 2 ; ; ; j j i ¡ m n u v ( ) i i ¼ k 2 u v X i ( ) 1 2 E j j 2 8 V = 1 u v 2 u u ; = i i 1 = Semidefinite Program

23. [ [ ] ] P P i i j r x r x x » » = = = u u u ; Ã ! k [ ] h i 8 8 k 6 V 1 i j i j 0 2 2 u u u = = X X i j 2 ; ; ; j j i ¡ m n u v ( ) i i ¼ k 2 u v X i ( ) 1 2 E j j 2 8 V = 1 u v 2 u u ; = i i 1 = 2 ( ) [ ] h i j j 8 k E i 0 · · 2 2 u v u v u ( ) i i i ¼ ; ; u v Semidefinite Program

24. u v v u k k 2 2 v 1 u 1 Intuition For each vertex, we have an orthogonal system of vectors. For adjacent vertices the vectors are close. Our goal is to pick one vector for each vertex. Green vectors correspond to vertex u. Red vectors correspond to vertex v.

25. Uniform Case

26. 1 p k ~ 2 j j ¢ u u = i i u = i k Intuition: Uniform Case • Assume all vectors have the same length: • Normalize vectors:

27. ~ ~ u u k i g ~ u j Algorithm: Uniform Case Pick a random Gaussian vector g.

28. h i Z ~ g u = i u ; i ~ ~ u u k i g ~ u j Algorithm: Uniform Case For each vertex u, project g on ui:

29. ( f ) = g k P S Z Z i 1 ¸ ¸ t t r : ~ ~ = = u u u i i u u k i g ~ u j t t Algorithm: Uniform Case For each vertex u, , where the threshold t is s.t.

30. = h b b l k d f S S S i i i i i i i 1 1 t t t 2 n c e w p r o a y ; e x p e c e s z e o s u u . ( f ) = g k P S Z Z i 1 ¸ ¸ t t r : ~ ~ = = u u u i i u u k i g ~ u j t t Algorithm: Uniform Case For each vertex u, , where the threshold t is s.t.

31. ~ ~ u u k i g ~ u j t t Algorithm: Uniform Case • Pick at random i from Suand assign xu = i.

32. p h i k Z ~ ~ ¢ u g u u = = i i i u ; i f g S Z i ¸ t : = u u i Recap • Denote • Pick a random Gaussian vector g. • Project g on ui: . • For each vertex u, . • Pick at random i from Suand assign xu = i. u

33. p h i k Z ~ ~ ¢ u g u u = = i i i u ; i f g S Z i ¸ t : = u u i Recap • Denote • Pick a random Gaussian vector g. • Project g on ui: . • For each vertex u, . • Pick at random i from Suand assign xu = i. u S S

34. p h i k Z ~ ~ ¢ u g u u = = i i i u ; i f g S Z i ¸ t : = u u i Recap • Denote • Pick a random Gaussian vector g. • Project g on ui: . • For each vertex u, . • Pick at random i from Suand assign xu = i. u S S

35. j j S S \ u v j j S S \ ¼ u v j j j j S S u v Uniform Case: Analysis • Assume, for brevity, uv = id. • What is the probability that the constraint for the edge (u, v) is satisfied? • Expected sizes of Su and Sv are 1. v u S S S S

36. ( ) h i Z Z Z Z S S i c o v u v ¸ ¸ t t = \ i i 2 S S u v ; ; : i i \ u v u v i i u v Expected size of . • if and • We know the joint ditribution of Zui and Zvi: each is a standard normal variable; their covariance is So we can compute the desired probability.

37. " " u v µ ¶ µ ¶ j j 1 S S ¡ 2 1 ¡ 2 \ " " u v u v ¼ k j j j j k S S u v Uniform Case: Analysis • The probability that the constraint for the edge (u, v) is satisfied: • Average over all constraints: v u S S S S

38. Non-uniform Case

39. [ ] P i r x » = u Non-uniform Case • Need to take lengths of vectors into account. • Should normalize vectors. • Cannot ignore lengths. • Solution: project each vector ui on several Gaussians; # of projections is proportional to |ui|2.

40. p ( ) l l k O 1 ¡ " o g n o g Conclusion and Open Problems Our algorithm can solve 1-to-d games [CMM’] New algorithm for Unique Games with approximation guarantee Prove or disprove the Unique Games Conjecture

41. Thank you! Questions?