Approximation Algorithms for Unique Games

1. Approximation Algorithms forUnique Games Konstantin Makarychev based on two papers with Eden Chlamtac Moses Charikar Yury Makarychev Princeton University

2. Unique Games colors

3. Unique Games Constraints colors

4. Unique Games colors

5. Unique Games colors Goal: Satisfy as many constraints as possible.

6. Example colors

7. Example colors

8. Example colors

9. 8 ( ) d k 2 1 3 3 + x x m o = 1 4 > > > ( ) d k 1 6 4 + < x x m o = 3 2 > : : : > > ( ) d k : 5 3 5 7 + x x m o = 1 9 Example: MAX 2LIN • Linear equations mod k, two var’s per equation. Maximize # of satisfied equations. variables = verticesequations = edges values= colors

10. 8 ( ) d k 2 1 3 3 + x x m o = 1 4 > > > ( ) d k 1 6 4 + < x x m o = 3 2 > : : : > > ( ) d k : 5 3 5 7 + x x m o = 1 9 Example: MAX 2LIN • Linear equations mod k, two var’s per equation. Maximize # of satisfied equations. variables = verticesequations = edges values= colors 99% of all equations is satisfiable

11. Unique Games Conjecture of Khot*03 • Given an instance where • 99% of all equations is satisfiable, • it is NP-hard to satisfy 1% of all equations.

12. Unique Games Conjecture • Used to prove best known hardness of approximation results for: MAX CUT, Min 2CNF Deletion MultiCut, Sparsest Cut, Vertex Cover and Graph Coloring

13. Unique Games Conjecture • Used to prove best known hardness of approximation results for: MAX CUT, Min 2CNF Deletion MultiCut, Sparsest Cut, Vertex Cover and Graph Coloring

14. Algorithms

15. Physical Interpretation vertices = particles colors = states adj. vertices interact with each other • If colors of adj. vertices satisfy the constraint, then • potential energy = 0 • else energy = 1

16. Physical Interpretation Find the state of the system with minimal energy E = 0 E = 0 E = 1 E = 1 E = 0 E = 0 Total energy = 2

17. Physical Interpretation Find the state of the system with minimal energy E = 0 E = 0 E = 1 E = 1 E = 0 E = 0 Idea: Use Quantum Mechanics!

18. Superposition of colorings Each vertex is colored simultaneously with several colors!

19. Superposition of colorings • The vertices are in quantum states! • Each state is described by a vector. • How can we get the “classical” solution?

20. Superposition of colorings • The vertices are in quantum states! • Each state is described by a vector. • How can we get the “classical” solution? Perform a measurement!

21. Semidefinite Programming

22. Approach Use Semidefinite Programming: • System of equation for vectors in a high dimensional space. • We can solve it in polynomial time. • Reconstruct a solution of the original problem – “Round the solution”.

23. Vector Configuration

24. 2 2 2 j j j j j j ¸ ¡ + ¡ ¡ u v v w u w i j j i m m à ! 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 = [ ] 8 k V i j 2 2 u v w m ; ; ; ; Semidefinite Program

25. Vector Configuration

26. Vector Configuration Find a subset of vectors S Violate few constraints- don’t separate close vectors Whp pick only 1 vector among orthogonal

27. ³ ´ ³ ´ p p l k O 1 ¡ l l k O 1 ¡ " o g " o g n o g = = l l k 1 1 0 c c o o g g n = 2 1 5 ( ) k O 1 = ( ) ¡ 2 1 ¡ ¡ ¡ ( ) l O " 1 " " k k ¡ " o g n Our results • (1- ) fraction of constraints is satisfied [GT’ 06] [K’ 02]

28. Can we do better? • No?Doing any better would disprove the Uniue Games Conjecture! [KKMO].

29. Methods • Probability • Functional Analysis • Geometry • Combinatorics • No Quantum Mechanics 

30. Thank you! Questions?