A Counterexample to Strong Parallel Repetition

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A Counterexample to Strong Parallel Repetition

• Ran Raz
• Weizmann Institute

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• Two Prover Games
• Player A gets x
• Player B gets y
• (x,y) 2R publicly known distribution
• Player A answers aA(x)
• Player B answers bB(y)
• They win if V(x,y,a,b)1
• (V is a publicly known function)
• Val(G) MaxA,B Prx,y V(x,y,a,b)1

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• Example
• Player A gets x 2R 1,2
• Player B gets y 2R 3,4
• A answers aA(x) 2 1,2,3,4
• B answers bB(y) 2 1,2,3,4
• They win if abx or aby
• Val(G) ½
• (protocol ax, b 2R 1,2)
• (alternatively by, a 2R 3,4)

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• Parallel Repetition
• A gets x (x1,..,xn)
• B gets y (y1,..,yn)
• (xi,yi) 2R the original distribution
• A answers a(a1,..,an) A(x)
• B answers b(b1,..,bn) B(y)
• V(x,y,a,b) 1 iff 8i V(xi,yi,ai,bi)1
• Val(Gn) MaxA,B Prx,y V(x,y,a,b)1

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• Parallel Repetition
• A gets x (x1,..,xn)
• B gets y (y1,..,yn)
• (xi,yi) 2R the original distribution
• A answers a(a1,..,an) A(x)
• B answers b(b1,..,bn) B(y)
• V(x,y,a,b) 1 iff 8i V(xi,yi,ai,bi)1
• Val(Gn) MaxA,B Prx,y V(x,y,a,b)1
• Val(G) Val(Gn) Val(G)n
• Is Val(Gn) Val(G)n ?

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• Example
• A gets x1,x2 2R 1,2
• B gets y1,y2 2R 3,4
• A answers a1,a2 2 1,2,3,4
• B answers b1,b2 2 1,2,3,4
• They win if 8i aibixi or aibiyi
• Val(G2) ½ Val(G)
• By a1x1, b1y2-2, a2x12, b2y2
• (they win iff x1y2-2)

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• Parallel Repetition Theorem R95
• 8G Val(G) lt 1 ) 9 w lt 1
• (s length of answers in G)
• Assume that Val(G) 1-?
• What can we say about w ?

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• Parallel Repetition Theorem
• Val(G) 1-? , (? lt ½) )
• R-95
• Hol-06
• For unique and projection games
• Rao-07
• (s length of answers in G)

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• Strong Parallel Repetition Problem
• Is the following true ?
• Val(G) 1-? , (? lt ½) )
• (for any game or for interesting special cases)
• Our Result G s.t. Val(G) 1-?,

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• Applications of Parallel Repetition
• 1) Communication Complexity direct product
  results PRW
• 2) Geometry understanding foams, tiling the
  space Rn FKO
• 3) Quantum Computation strong EPR paradoxes
  CHTW
• 4) Hardness of Approximation BGS,Has,Fei,K
  ho,...

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• EPR Paradox 9 G s.t.
• ValQ(G) gt Val(G)
• ValQ(G) value when the provers
• share entangled quantum states
• CHTW 04 9 G s.t.
• ValQ(G) 1 and Val(G) 1-?
• (for some constant ? gt 0)
• Using Parallel Repetition 9 G s.t.
• ValQ(G) 1 and Val(G) ?
• (for any constant ? gt 0)

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• PCP Theorem BFL,FGLSS,AS,ALMSS
• Given G (with constant answer size)
• It is NP hard to distinguish between
• Val(G) 1 and Val(G) 1-?
• (for some constant ? gt 0)
• Using Parallel Repetition
• It is NP hard to distinguish between
• Val(G) 1 and Val(G) ?
• (for any constant ? gt 0)

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• Unique Games (UG)
• G is a UG if V(x,y,a,b) satisfies
• 8 x,y,a 9 unique b, V(x,y,a,b) 1
• 8 x,y,b 9 unique a, V(x,y,a,b) 1
• Unique Games Conjecture Khot
• 8 constant ? gt 0, 9 constant s, s.t.
• Given a UG G (with answer size s)
• It is NP hard to distinguish between
• Val(G) 1-? and Val(G) ?

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• UGC and Max-Cut KKMO
• UGC ) 8 ? gt 0, given a graph G,
• It is NP hard to distinguish between
• Max-Cut(G) 1-?2 and
• Max-Cut(G) 1-2?/p
• Using Strong Parallel Repetition
• UGC , 8 ? gt 0, given a graph G,
• It is NP hard to distinguish between
• Max-Cut(G) 1-?2 and
• Max-Cut(G) 1-2?/p

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• Odd Cycle Game CHTW,FKO
• A gets x 2R 1,..,m (m is odd)
• B gets y 2R x,x-1,x1 (mod m)
• A answers aA(x) 2 0,1
• B answers bB(y) 2 0,1
• They win if xy , ab

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• Odd Cycle Game CHTW,FKO
• A gets x 2R 1,..,m (m is odd)
• B gets y 2R x,x-1,x1 (mod m)
• A answers aA(x) 2 0,1
• B answers bB(y) 2 0,1
• They win if xy , ab

1
0
0
1
1
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• Parallel Repetition of OCG
• A gets x1,..,xn 2R 1,..,m
• B gets y1,..,yn 2R 1,..,m
• 8 i yi 2R xi,xi-1,xi1 (mod m)
• A answers a1,..,an 2 0,1
• B answers b1,..,bn 2 0,1
• They win if 8 i xiyi , aibi

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• Parallel Repetition of OCG
• A gets x1,..,xn 2R 1,..,m
• B gets y1,..,yn 2R 1,..,m
• 8 i yi 2R xi,xi-1,xi1 (mod m)
• A answers a1,..,an 2 0,1
• B answers b1,..,bn 2 0,1
• They win if 8 i xiyi , aibi
• Motivation FKO Max-Cut vs. UGC,
• Understanding foams, Tiling the space

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• Our Results
• (match an upper bound of FKO)
• For n ¼ m2,
• For n ?(m2),

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• Odd Cycle Game
• A gets x, B gets y. If they can
• agree on an edge e that doesnt
• touch x,y, they win !

0
1
1
0
1
0
1
0
1
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• Parallel Repetition of OCG
• A gets x1,..,xn, B gets y1,..,yn.
• If they can agree on edges e1,..,en
• that dont touch x1,..,xn, y1,..,yn,
• they win !

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• Holensteins Lemma B,KT
• A has f W ! R, B has g W ! R,
• s.t., f-g1 O(?)
• Using shared randomness, A can
• choose u 2f W, and B v 2g W,
• s.t. Probuv 1-O(?)

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• Distribution P
• m2k1, P-k,k ! R (symmetric)
• 1) P(i) ¼ (k1-i)2 / m3
• 2) P(0) (1/m)
• 3) P(k) P(-k) (1/m3) (negligible)

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• Distribution P
• m2k1, P-k,k ! R (symmetric)
• 1) P(0) (1/m)
• 2) P(k) P(-k) (1/m3) (negligible)
• 3)

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• Distributions on the Odd Cycle
• E Edges of the odd cycle.
• Given x, A defines fx E ! R fxP,
• concentrated on the edge opposite to x
• Given y, B defines gy E ! R gyP,
• concentrated on the edge opposite to y
• fx ¼ gy (since x,y are adjacent)
• fx-gy1 O(1/m)

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• Holensteins Lemma
• A has f W ! R, B has g W ! R,
• s.t., f-g1 O(?)
• Using shared randomness, A can
• choose u 2f W, and B v 2g W,
• s.t. Probuv 1-O(?)
• OCG Using fx,gy, A,B can agree on
• an edge e that doesnt touch x,y,
• with probability 1-O(1/m)

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• Our Protocol
• Given x(x1,..,xn), A defines fx En ! R,
• Given y(y1,..,yn), B defines gy En ! R,
• Lemma
• Using Holensteins lemma, A,B agree on
• edges e1,..,en that dont touch x1,..,xn,
• y1,..,yn, with probability

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• Proof Idea
• Typically in coordinates
• and in
• coordinates
• n/3 coordinates cancel each other. We
• are left with distance

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• Proof Idea
• Hence, typically

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• Follow Up Works
• 1) Generalizations to unique games
• BHHRRS Protocols for parallel
• repetition of any unique game
• 2) Tiling the space Rn KORW,AK
• Rn can be tiled (with translations in Zn),
• by objects with surface area similar to
• the one of the sphere (with volume 1)

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The End