1. Unique Games Approximation Amit Weinstein Complexity Seminar, Fall 2006 Based on: “Near Optimal Algorithms for Unique Games" by M. Charikar, K. Makarychev, Y. Makarychev

2. 2 Outline What is Unique Game?  Definition  Solving a Satisfiable Game  Generalization: d-to-d games Known Hardness and Approximation results Integer Programming and SDP representation Rounding Algorithm  How is it done  What does it guarantee

3. 3 What is Unique Game? A Constraints Graph k – Domain size Objective: Satisfy as many edges as possible

4. 4 MaxCut as a Unique Game

5. 5 Can we solve a satisfiable game? Greedy ! Go over all possible x’s Complete the assignment Check Solution

6. 6 Generalization A game is called d-to-d if:  For each edge (u,v)  Given an assignment to v  Only d possible assignments to u will satisfy this edge So what is a Unique Game?  A 1-to-1 game Can you think of a simple 2-to-2 game?  3-Coloring Can we solve a 2-to-2 satisfiable game?

7. 7 Known Approximations (and bounds) General Unique Game  Approx. 1/k (Random Assignment) MaxCut:  Approx. of 0.878… using SDP relaxation  NP-hard to approx Hastad 02 2LinEq GF2  Approx. 1/2 (Random Assignment)  NP-hard to approx Hastad 02 can be very small Geomans, Williamson 95

8. 8 Unique Games Conjecture (UGC) This is the main Conjecture of Unique Games  Still haven’t been proven  Most people assume it is true YES INSTANCE At least of the edges can be satisfied NO INSTANCE At most of the edges can be satisfied

9. 9 Assuming the UGC is true MaxCut  We know approximation 0.878…  It is NP-hard to approx. within any factor Khot, Kindler, Mossel, O’Donnel 04  Again, this means 0.878… is optimal Vertex Cover  We know approximation 2  It is NP-hard to approx. within any factor Khot, Regev 03  Meaning 2 is optimal

10. 10 Known Unique Game Approx. Results: This Article: Meaning

11. 11 Unique Game as Integer Programming We define: Claim: And therefore

12. 12 Integer Programming – Edges weight Proof for: 00100 10000 01000

13. 13 Unique Game as Integer Programming Remember: The program:

14. 14 From Integer Programming to SDP Discrete variables to vectors We also add a few constraints We don’t need Triangle Inequalities on the norms From now on, all variables u i are vectors !

15. 15 SDP – Some Intuition Size = probability Direction = correlation  Small angle – correlated  Large angle – uncorrelated  Reminder:

16. 16 SDP – Solution Illustration

17. 17 Rounding Algorithm – The Idea For simplicity, we assume We pick a random Gaussian vector g  Each coordinate of g

18. 18 Rounding Algorithm – The Idea Define the Sets:  Possible Values  Choose a threshold s.t.  Randomly choose from these Sets What is

19. 19 Rounding Alg. – The Idea Calculation Chosen Independent Sum over all possible i By Definition

20. 20 Rounding Alg. – The Idea Calc. cont.. By our choice: Since there are k such possibilities: From the Promise and assumptions For intuition, Not accurate

21. 21 What about our assumptions ? Lengths assumption Distance assumption We repeat the procedure  #times ~ vector’s length  For vector u i we repeat times  Using different random vectors We choose k random Gaussian vectors Starting here

22. 22 The Rounding Algorithm Define Recall: Define The Assignment: We now need to analyze it Ignore empty Sets

23. 23 SDP – Rounding Illustration

24. 24 Rounding Algorithm – Definitions The distance between two vertices: Also, Which basically holds:  When is the angle between them  If one of the vectors is 0, we set

25. 25 Rounding Algorithm – Definitions We define a measure Notice:

26. 26 Rounding Algorithm – Proof Sketch 3 steps, similar to the easy case Lemma 3.3: Bound Lemma 3.7: Bound Averaging

27. 27 Rounding Algorithm – Lemma 3.3 We define:

28. 28 Rounding Alg. – Lemma 3.3 Proof. Appendix Lemma B.3 Appendix Lemma B.1

29. 29 Rounding Alg. – Lemma 3.3 Proof We get By Definition

30. 30 Rounding Algorithm – Proof Sketch 3 steps, similar to the easy case Lemma 3.3: Bound Lemma 3.7: Bound Averaging 

31. 31 Rounding Algorithm – Lemma 3.7. Proof: Our measure properties: Lemma 3.3

32. 32 Rounding Alg. – Lemma 3.7 Proof Consider For any We know: So by Markov inequality: Out measure properties

33. 33 Rounding Alg. – Lemma 3.7 Proof The function is convex at [0,1] By Jensen’s inequality: Allows us to insert the Sum into the function

34. 34 Rounding Alg. – Lemma 3.7 Proof The function is convex at [0,1] By Jensen’s inequality: Allows us to insert the Sum into the function

35. 35 Rounding Algorithm – Proof Sketch 3 steps, similar to the easy case Lemma 3.3: Bound Lemma 3.7: Bound Averaging  

36. 36 Rounding Algorithm – The Result There is a polynomial time algorithm (which we saw), that find an assignment which satisfies given the optimal assignment satisfies at least of the constraints.

37. 37 Rounding Alg. – The Result ’ s Proof We consider only For So So averaging over all, using Jensen and the convexity of we get:  Again, we insert the average sum inside.

38. 38 Rounding Algorithm – Proof Sketch 3 steps, similar to the easy case Lemma 3.3: Bound Lemma 3.7: Bound Averaging   

39. 39 Proof Meaning Given SDP solution better than We found an assignment We proved it satisfies This is what we wanted

40. 40 Summary Given a Unique Game Input Defined Integer Programming Translated into SDP Used a rounding Algorithm We showed that if at least could be satisfied Our solution will give:

41. 41 Questions ? ????? ?????????????????????????????????????

42. 42 Rounding Algorithm – Filling Holes Lemma: We use:

43. 43 Rounding Algorithm – Filling Holes Lemma: W.l.o.g. assume

44. 44 Normal Gaussian vectors properties back