Unique Games Approximation

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

4. MaxCut as a Unique Game

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

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. can be very small 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 • 2LinEqGF2 • Approx. 1/2 (Random Assignment) • NP-hard to approx Hastad 02 Geomans, Williamson 95

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

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

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

12. 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 Integer Programming – Edges weight • Proof for:

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

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

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

16. SDP – Solution Illustration

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

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

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

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

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

23. SDP – Rounding Illustration

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

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

27. Rounding Algorithm – Lemma 3.3 • We define:

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

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

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

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

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

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

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

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

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

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. Questions ? ?? ?? ?? ?? ? ? ??? ??? ? ?

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

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

44. Normal Gaussian vectors properties back