On Parallel Repetition

1. Thomas Holenstein Microsoft Research, SVC Dagstuhl 15.9.2008 On Parallel Repetition

2. Parallel Repetition • Problem Statement and Connection to PCP • Consistent Sampling Lemma • Raz’s Counterexample [Raz 08] • Parallel Repetition Theorem

3. Problem Statement and Theorem i=1,…,n: (xi,yi) ← PXY (x,y) ← PXY x1,…,xn x y1,…,yn y a1,…,an a b1,…,bn b For all i: Q(xi,yi,ai,bi)? Q(x,y,a,b)? Does this decrease the winning probability? Theorem[Raz 95, H 07]:Yes, from 1 - γ to (1 - O(γ3))n/log(|A||B|) Why would I care about this? Why is this not trivial? Theorem[Raz 08]:G with v(G)= 1 – γ, v(Gn)=(1-O(γ2))n Players may not answer independently. (1 – γ)n not true!

4. The PCP Theorem (strong version) 7 8 It is NP-hard to satisfy –+ of the clauses of a satisfiable 3SAT formula [Håstad 01] Proof sketch: 3-SAT is NP-hard [Cook71, Levin73] Gap AmplificationIt is NP-hard to satisfy 99% of the clauses of a satisfiable 3-SAT formula [BFL91, BFLS91, FGLSS 91, AS92, ALMSS92, Dinur07] Parallel Repetition It is NP-hard to win in a game as before with probability  [Raz98, H] Fourier Techniques It is NP-hard to satisfy 7/8+  of the clauses of a 3-SAT formula

5. SAT-formulas 1 2 207 921 (x1x2x3)  (x1x5x9)    (x2x6x9)  …  (x16x18x41) Pick a random clause, send clause to Alice, Variable to Bob 9 207 2 0 (1,0,0) Fraction of unsatisfied clauses = O(rejection probability)

6. Overview • Problem Statement and connection to PCP • Consistent Sampling Lemma • Raz’s Counterexample [Raz 08] • Parallel Repetition Theorem

7. Consistent Sampling Subtask: Alice gets PS, Bob gets PS’, minimize Pr[SS’] x1,…,xn y1,…,yn 1 4 a1,…,an b1,…,bn 1 ½ 2 3 A s4 5 B s3 6 0 s2 s4 s1 s3

8. Overview • Problem Statement and connection to PCP • Consistent Sampling Lemma • Raz’s Counterexample [Raz 08] • Parallel Repetition Theorem

9. A Counterexample to Strong Parallel Repetition [Raz, FOCS 08]  G: v(G) = 1- and v(Gn) = (1-O(2))n v(G) = 1 – O(1/k) v(Gn)  ½ for n = O(k2) Odd cycle game (2k+1): x  {1, ..., 2k+1}, y  {x-1,x,x+1} Q(x,y,a,b)  (x = y) = (a = b)

10. A Counterexample to Strong Parallel Repetition [Raz, FOCS 08] Start with a single instance (2k+1 = 11): 1 2 3 4 5 6 7 8 9 10 11 Pick an edge with the consistent sampling lemma. x=7 y=8 0 1 0 0 1 0 1 0 1 0 1 Parameters are such that: ||PS - PT|| = O(1/k), Pr[Bad edge] = O(1/k2) ||PSn- PTn|| = O(1) for n = O(k2) QED!

11. Overview • Problem Statement and connection to PCP • Consistent Sampling Lemma • Raz’s Counterexample [Raz 08] • Parallel Repetition Theorem

12. Parallel Repetition Theorem i=1,…,n: (xi,yi) ← PXY x1,…,xn y1,…,yn a1,…,an b1,…,bn v(G) = 1 - γ implies v(Gn)  (1-O(γ3))n/log(|A||B|)

13. Decomposition Goal: show Pr[W1  W2  ... Wn] << v x1,…,xn y1,…,yn Pr[W1  ... Wn] = Pr[W1] Pr[W2|W1] ... Pr[Wn|W1  ... Wn-1] a1,…,an b1,…,bn It is sufficient to show that Pr[Wm+1|W1  ... Wm] ≤ v + ε Also enough: for all i1, …,im there exists a j such that Pr[Wj|Wi1  ... Wim] ≤ v + ε ε =O( log(|A||B|) – log(Pr[Wi1  ... Wim]) mn Increasing n decreases ε!

14. Proof Given: strategy of Alice and Bob, indices i1,…,imGoal: show that for some j: Pr[Wj|Wi1  ... Wim] ≤ v + ε x1,…,xn y1,…,yn Assume otherwise and get a strategy for the initial game. a1,…,an b1,…,bn x y Can only work if: • xj = x and yj = y x1,…,xn y1,…,yn • (x1,…,xn,y1,…,yn) have correct distribution aj bj a1,…,an b1,…,bn

15. Embedding x1,…,xn y1,…,yn PXY X Y a1,…,an b1,…,bn Goal: embed into (X,S) PS,T|X=x,Y=y Goal: embed into (Y,T) I know of two sufficient conditions: • S ↔ X ↔ Y ↔ T • S = T and PS|X=x,Y=y ≈ PS|X=x ≈ PS|Y=y

16. Main Lemma Assume: Pr[W5 | W1 W2] is big x1,…,xn y1,…,yn x y Won games Won games a1,…,an b1,…,bn x1 x2 x3 x4 x x6 x7 x1 x2 x3 x4 x6 y1 y2 y7 y1 y2 y3 y4 y y6 y7 a5 a1 a2 a1 a2 b5 b1 b2 b1 b2 Send the corresponding answer to the referee Apply the given strategy for multiple games Forget part of the generated information Using PS|X=xY=y ≈ PS|X=x ≈ PS|Y=y Using S ↔ X ↔ Y ↔ T a1 a2 a3 a4 a6 a7 b1 b2 b3 b4 b6 b7

17. Overview v(G) = 1 – γ v(Gn)  (1 – O(γ3))n/log(|A||B|) [Raz 98,H07] v(P) = 1 – γ v(Pn)  (1 – O(γ2))n [Rao 08] G: v(G) = 1 – γ and v(Gn)  (1 – O(γ2))n [Raz 08] ~ G: v(G) = ¾ and v(Gn)  (½)n/O(log(|A| |B|)) [FV 02] • Connection with Foams: [FKO 07, KORW 08] Parallel Repetition of Unique Games: [BHHRRS ’08] Thank you!