Rounding Parallel Repetitions of Unique Games

1. Rounding Parallel Repetitions of Unique Games Boaz Barak Moritz Hardt IshayHaviv AnupRao OdedRegev David Steurer TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAA

2. XOR Games Alice Bob • The referee chooses (u, v, c) from some known distribution • He sends u to Alice and v to Bob • Each player responds with a bit in {-1,1} (no communication allowed) • The players win iffa¢b = c • The value of a game is the maximum success probability the players can achieve, and is denoted by val(G) • The players can use shared randomness (but it can’t help them) v u a b win Referee lose

3. 5 6 1 3 4 3 2 1 6 5 4 2 Example: an even cycle game • Alice’s question u is a uniformly chosen vertex • Bob’s question v is • u w.p. ½ • u+1 w.p. ¼ • u-1 w.p. ¼ • Their goal is to give the sameanswer iffthey have the same vertex • val(G) = 1 1 5 4 -1 

4. 1 3 5 1 2 2 4 3 4 5 Example: an odd cycle game • val(G) = 9/10 1 5 4 1 

5. 1 1 1 4 4 4 3 3 3 5 5 5 2 2 2 Parallel Repetition • The referee chooses (u1, v1, c1), (u2, v2, c2),…,(uℓ, vℓ, cℓ) indep. • He sends u1,…,uℓ to Alice and v1,…,vℓ to Bob • Each player responds with ℓbits • The players win iff for all i, ai¢bi = ci • Denote val(G)=1-² • Clearly, val(Gℓ)¸(1-²)ℓ • Is val(Gℓ)=(1-²)ℓ ? • NO! • Known upper bounds (‘parallel repetition theorems’): • val(Gℓ)·(1-²32)(ℓ) (for general games) [Raz95] • val(Gℓ)·(1-²3)(ℓ) (for general games) [Holenstein06] • val(Gℓ)·(1-²2)(ℓ) (for projection games) [Rao07] • Strong parallel repetition conjecture: [FeigeKindlerO’Donnell07] • val(Gℓ)·(1-²)(ℓ) • [Raz08]: The conjecture is false! • The odd cycle game satisfies val(Gℓ)¸(1-²2)O(ℓ) (for large enough ℓ)

6. SDP Relaxation of XOR games[FeigeLovász92, GoemansWilliamson94] • Define • Clearly sdpval(G)¸val(G) • Inequality can be strict: for odd cycle game, val=1-1/2m but sdpval=1-O(1/m2) • Fact [FeigeLovasz92,MittalSzegedy07]: sdpval(Gℓ)=sdpval(G) ℓ • Cor: If sdpval(G)=1- then val(Gℓ)·(1-)ℓ

7. Our Results • Thm 1: For any XOR game with sdpval(G)=1-, • Cor: For any XOR game with sdpval(G)=1- and large enough ℓ, • This shows that when taking parallel repetitions, the value of the game behaves like the SDP relaxation • This also gives an efficient way to approximate the asymptotic value of XOR games • Thm 2: For any unique game with alphabet size k and sdpval(G)=1-,

8. Motivation: UGC • In a unique game, the answers a, b are from some alphabet [k], and the constraint is of the form ¼(a)=b where ¼ is a permutation [CaiCondonLipton90, FeigeLovász92]. • Unique Games Conjecture[Khot02]: ,>0 k such that it is NP-hard to determine whether a given unique game G with alphabet [k] has val(G)1- or val(G)< • Implies lots of tight hardness results (MaxCut[KhotKindlerMosselO’Donnell04], VertexCover[KhotRegev03], …) • For general games such a statement is true (even with =0) • The proof is by combining the PCP theorem with (Raz’s) parallel repetition theorem • Can we prove the UGC using a similar strategy? • Maybe, but difficult due to relatively good approximation algorithms for the value of unique games [Trevisan05, CharikarMakarychevM06, GuptaTalwar06, ChlamtacMakarychevM06] and the counterexamples for strong parallel repetition

9. The Proof

10. Outline • Preliminaries: Hellinger distance • Preliminaries: Correlated sampling lemma • Proof of Thm 1: Strategy for repeated games

11. Statistical Distance of Repetitions • Consider the two distributions • D1: a coin with head probability 1 • D2: a coin with head probability 1-² • Their statistical distance is ¢(D1,D2)=² • Q: How big should k be for ¢(D1k,D2k) to be (1)? • A: 1/²

12. Statistical Distance of Repetitions • Now consider • D1: a coin with head probability (1+²)/2 • D2: a coin with head probability (1-²)/2 • Their statistical distance is again ¢(D1,D2)=² • Q: How big should k be for ¢(D1k,D2k) to be (1)? • A: 1/²2 • Why??

13. The Hellinger Distance • Def: The Hellinger distance between two discrete distributions P=(p1,...,pn) and Q=(q1,…,qn) is • Lemma 1: • Lemma 2: • Cor:

14. Correlated Sampling • Alice is given a distribution P and Bob is given a distribution Q • Each is supposed to output a sample from her/his given distribution (using shared randomness) • Goal: maximize probability of equal outputs • Lemma [Broder97,KleinbergTardos99,Holenstein07]: This can be done with success probability ¸1-2¢(P,Q)

15. Correlated Sampling - Proof • Assume for simplicity each distribution is uniform over some set

16. Correlated Sampling - Analysis S • Probability of success is: • For general distributions, one uses the ‘area under the function’ T

17. Proof of Thm 1 • Thm 1: For any XOR game with sdpval(G)=1-, • Proof: • Let xu be the unit vectors in the optimal solution, so • Our goal is to show a strategy for the repeat game Gℓ with success probability 1-O(ℓ)

18. Attempt 1: Strategy for G • The strategy: • Alice and Bob choose a unit vector w uniformly • On input vector xu, they output sign(hxu,wi) • Analysis: If c=1 (equality constraint), the failure probability is arccos(hxu,xvi)¼ if hxu,xvi=1-. Similarly for c=-1.

19. Attempt 2: Strategy for G • The strategy: • Using correlated sampling, the players choose a vector w by using a distribution centered around their input vector • On input vector xu, they output sign(hxu,wi)

20. Attempt 2: Strategy for G • Analysis: • Assume c=1 and hxu,xvi=1- • The probability that w falls in the bad area is only ¼ • But the probability of agreeing on the same w is 1-2¢(P,Q) and ¢(P,Q) ¼

21. Strategy for Gℓ • Given ℓ vectors xu1,…,xuℓ, the players choose an ℓ-tuple (w1,…,wℓ) using correlated sampling from the product distribution of the previous distributions • The probability that any of the w’s falls in the bad area is only ¼ℓ • Because H2(P,Q)¼ we have that H2(Pℓ,Qℓ)¼ℓ and so the probability of them not agreeing on the same tuple of w’s is only (ℓ) • The overall success probability is ¼1-ℓ, as required

22. Conclusions • We showed that when taking parallel repetitions of XOR games, the value behaves like the SDP value • In the paper we • extend this to unique games (not true for general games), and • derive bounds for specific games • Open questions: • Get rid of the log(1/) in Theorem 2 • Should follow from a different choice of distribution, but analysis seems hard • Other applications of the Hellinger distance?