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Understanding Parallel Repetition Requires Understanding Foams

Understanding Parallel Repetition Requires Understanding Foams. Uri Feige Microsoft. Guy Kindler Weizmann. Ryan O’Donnell CMU. What we wanted to solve. Strong Parallel Repetition Problem: Let G be a 2-prover 1-round game with answer sets A , B .

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Understanding Parallel Repetition Requires Understanding Foams

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  1. Understanding Parallel RepetitionRequires Understanding Foams Uri FeigeMicrosoft Guy KindlerWeizmann Ryan O’DonnellCMU

  2. What we wanted to solve Strong Parallel Repetition Problem: Let G be a 2-prover 1-round game with answer sets A, B. Is it true that val(G ) ·1 − ) val(Gd) · (1 −())d/log(|A||B|) ?

  3. A special case Strong Unique-Games Parallel Repetition Problem: Let G be a 2P1R game with answer sets A, B and unique constraints. Is it true that val(G ) ·1 − ) val(Gd) · (1 −())d/log(|A||B|) ?

  4. A further special case Strong 2-Lin Parallel Repetition Problem: Let G be a 2P1R game with 2-Lin constraints. Is it true that val(G ) ·1 − ) val(Gd) · (1 −())d?

  5. A further further special case Odd-Cycle Parallel Repetition Problem: Let GCm be the Odd-Cycle game of length m, which satisfies Is it true that val(GCm) = 1 − (1/m). Is it true that val(GCmd) ·(1 −(1/m))d?

  6. Further reduces to Torus Blocking Problem on (Zmd)1: Let (Zmd)1 be the “discrete torus graph”: vertex set = Zmd, edge set = {(x, y) : ||x−y||1· 1}. To block all cycles that “wrap around”, what’s the least fraction of edges you can delete?

  7. Our results • Improved lower bound for Torus Blocking Problem, which implies • Improved upper bounds for Odd Cycle Parallel Repetition problem. • At least, if you look at the parameters in the right way.

  8. This looks kind of pathetic

  9. But it’s not our fault

  10. Further further reduces to Foam on Rd / Zd Problem: What is the least surface area of a cell which tiles Rd by Zd ?

  11. Further further reduces to Foam on Rd / Zd Problem: What is the least surface area of a cell which tiles Rd by Zd ?

  12. Kelvin foam

  13. Similar questions are hard open problems in geometry

  14. Foam on Rd / Zd LetA(d)denote the least possible surface area… Upper bound?  A(d) ·d. Lower bound?  ÷ 2. the unit cube the volume-1 ball

  15. Other bounds • A(d) · d− 2−O(d log d) (put a radius-½ sphere at cube’s corner) • (the hexagon was optimal [Choe’89]) • For d = 3, nothing known except sphere vs. cube: 2.42 ¼ (9/2)1/3·A(3) < 3.Experts’ d = 3 conjecture: same combinatorial structure as “Kelvin Foam”

  16. A prize For £100: Prove or disprove: A(d) ¸d1−o(1). For £25:Prove

  17. Foams as torus blockers Take the unit cube in Rd. Identify opp. faces so it’s a torus.

  18. Foams as torus blockers Take the unit cube in Rd. Identify opp. faces so it’s a torus. To block all cycles that “wrap around”, what’s the least amount of “wall” (d −1 dimensional surface) you need to build?

  19. Foams as torus blockers Take the unit cube in Rd. Identify opp. faces so it’s a torus. To block all cycles that “wrap around”, what’s the least amount of “wall” (d −1 dimensional surface) you need to build? (Hence the ÷ 2: surface counted twice – inside and outside.)

  20. A worse lower bound: ssss • Wall S at least blocks all axis-parallel cycles. • So projecting S onto d faces must cover them. • Let P be a tiny patch on S, with unit normal n. • Area contributed to projection on ith face: |hn, eii| area(P) • Sum over i: Equals (i |ni|)· area(P) ·· area(P) [Cauchy-Schwarz] • Integrate over P: · · area(S). • But this contribution better exceed d. n P

  21. A worse lower bound: ssss • Wall S at least blocks all axis-parallel cycles. • So projecting S onto d faces must cover them. • Let P be a tiny patch on S, with unit normal n. • Area contributed to projection on ith face: |hn, eii| area(P) • Sum over i: At most hn, (1, …, 1)i area(P) ·· area(P) [Cauchy-Schwarz] • Integrate over P: · · area(S). • But this contribution better exceed d. We already lost here. n P

  22. What’s this got to do with Parallel Repetition?What is Parallel Repetition?

  23. 1 2 3 4 5 6 7 8 9 10 Bipartite Constraint Graphs Label Set = { } Y X 11 12 w – a weight 13 – a constraint 14 The w’s sum up to 1. 15 : not OK 16 OK OK 17 OK not OK 18 OK OK 19 OK not OK 20 Whole thing is called G. val( G ) denotes max weight simultaneously satisfiable.

  24. 1 2 3 4 5 6 7 8 9 10 Bipartite Constraint Graphs 2-Prover 1-Round Games in complexity theory Label Set = { } Y X 11 12 w – a weight 13 – a constraint 14 The w’s sum up to 1. 15 : not OK 16 OK OK 17 OK not OK 18 OK OK 19 OK not OK 20 Whole thing is called G. val( G ) denotes max weight simultaneously satisfiable.

  25. 1 2 3 4 5 6 7 8 9 10 Bipartite Constraint Graphs Nonlocal Games in foundations of quantum mechanics Label Set = { } Y X 11 12 w – a weight 13 – a constraint 14 The w’s sum up to 1. 15 : not OK 16 OK OK 17 OK not OK 18 OK OK 19 OK not OK 20 Whole thing is called G. val( G ) denotes max weight simultaneously satisfiable.

  26. 1 2 3 4 5 6 7 8 9 10 Parallel Repetition: d “rounds” Label Set = { } Y X 11 12 w – a weight 13 – a constraint 14 The w’s sum up to 1. 15 : not OK 16 OK OK 17 OK not OK 18 OK OK 19 OK not OK 20 Whole thing is called G. val( G ) denotes max weight simultaneously satisfiable.

  27. Parallel Repetition: d “rounds” d Label Set = { } Yd Xd w – a weight – a constraint 1 8 4 3 8 14 20 13 17 18 The w’s sum up to 1. : not OK eg: OK OK OK weight = w1,14 w8,20 w4,13 w3,17 w8,18 not OK OK OK constraint = 1,148,20 4,13 3,17 8,18 OK not OK d d Whole thing is called G. val( G ) denotes max weight simultaneously satisfiable.

  28. Value under Parallel Repetition True or False? val(Gd) = val(G)d? val(Gd) · val(G)? val(G2) < val(G)? val(Gd) ! 0 as d!1 ? false true false true (took 6 years to prove)

  29. Raz’s Parallel Repetition Theorem Raz ’95: val(G ) ·1 −) val(Gd) · (1 −poly()) d/log(# labels) Tremendously important theorem for proving hardness of approximation results. Holenstein ’07: poly() can be 3 / 4000. Strong Parallel Repetition Problem: can this be improved to ()?

  30. The “2-Lin” special case # labels = 2, each constraint is either “=” or “” Feige-Lovász ’91 + Goemans-Williamson ’95: val(G ) ·1 −) val(Gd) · (1 −c2)) d, where c = 2/4. Strong 2-Lin Parallel Repetition Problem: Can this be improved to ()? My conjecture: Yes. My motivation: Would show that sharp hardness-of-approx for Max-Cut is “Unique Games Conjecture”-complete, not just “Unique Games Conjecture”-hard.

  31. Simplest 2-Lins: The Odd Cycle Games                m nodes ) val = 1 – 1/m

  32. Simplest 2-Lins: The Odd Cycle Games = =  = =   =  =   = =   = =     = =   = = = 1/3 total weight on self-loops ) val = 1 – (2/3)/m

  33. After Parallel Rep: Discrete Torus Graph Z52 (x,y) an edge iff ||x-y||1· 1 Constraints 1st col. diff., 2nd col. same 1st col. same, 2nd col. diff. 1st col. diff., 2nd col. diff. 1st col. same, 2nd col. same (self-loops, not pictured) NB: Constraints are “unique”

  34. After Parallel Rep: Discrete Torus Graph Z52 (x,y) an edge iff ||x-y||1· 1 Constraints 1st col. diff., 2nd col. same 1st col. same, 2nd col. diff. 1st col. diff., 2nd col. diff. 1st col. same, 2nd col. same (self-loops, not pictured) NB: Constraints are “unique”

  35. After Parallel Rep: Discrete Torus Graph Z52 (x,y) an edge iff ||x-y||1· 1 Given set of Failure Edges, there’s a corresp. labeling iff all “topologically nontrivial” cycles blocked(*) NB: Constraints are “unique”

  36. val(GCmd) vs. Torus Blocking Basically(*), val(GCmd) = 1 −(d, m), where (d, m) = least fraction of edges you need to delete from Zmd graph to eliminate all cycles that “wrap around”. To prove strong upper bound for val(GCmd), must prove strong lower bound for (d, m).

  37. Discrete vs. Continuous Foams But strong lower bound for (d, m) implies strong lower bound for A(d). Proposition: Upper bound for A(d) implies upper bound for (d, m). Specifically, (d, m) ·const. A(d) / m. Proof:

  38. Discrete vs. Continuous Foams But strong lower bound for (d, m) implies strong lower bound for A(d). Proposition: Upper bound for A(d) implies upper bound for (d, m). Specifically, (d, m) ·const. A(d) / m. Proof:

  39. Hence the paper’s title To understand the truth about parallel repetition, you must get good upper bounds for val(GCmd) (a special case of a special case of a special case of the general case). But this requires good lower bounds for the continuous Rd / Zd Foam Problem.

  40. Our results What do we actually prove in the paper?! Main Theorem: The continuous foam lower bound can be discretified into a lower bound for (d, m): (d, m) ¸(if d·m2 log m, say). Hence val(GCmd) · 1 − Proof: A lot of Fourier analysis.

  41. Our results What we got: val(GCmd) · 1 − Best previously: 1 −(d) ¢ (1/m)2 What we really wanted: 1 −(d) ¢ (1/m) m = 33

  42. The End (for now)

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