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Uri Feige Microsoft Understanding Parallel Repetition Requires Understanding Foams Guy Kindler Weizmann Ryan ODonnell CMU

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

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

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

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

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

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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.

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This looks kind of pathetic

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But its not our fault

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

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

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Kelvin foam

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Similar questions are hard open problems in geometry

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

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Other bounds A(d) · d 2 O(d log d) (put a radius-½ sphere at cubes corner) (the hexagon was optimal [Choe89]) 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

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

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Foams as torus blockers Take the unit cube in d. Identify opp. faces so its a torus.

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Foams as torus blockers Take the unit cube in d. Identify opp. faces so its a torus. To block all cycles that wrap around, whats the least amount of wall (d 1 dimensional surface) you need to build?

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

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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: | h n, e i i | area(P) Sum over i: Equals ( i |n i |) · area(P) · · area(P) [Cauchy-Schwarz] Integrate over P: · · area(S). But this contribution better exceed d. P n

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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: | h n, e i i | area(P) Sum over i: At most h n, (1, …, 1) i area(P) · · area(P) [Cauchy-Schwarz] Integrate over P: · · area(S). But this contribution better exceed d. P n We already lost here.

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Whats this got to do with Parallel Repetition? What is Parallel Repetition?

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Bipartite Constraint Graphs w – a weight Label Set = { } – a constraint not OK OK not OK OK not OK The ws sum up to 1. Whole thing is called. val( ) denotes max weight simultaneously satisfiable. X Y

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Bipartite Constraint Graphs w – a weight Label Set = { } – a constraint not OK OK not OK OK not OK The ws sum up to 1. Whole thing is called. val( ) denotes max weight simultaneously satisfiable. X Y Prover 1-Round Games in complexity theory

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Bipartite Constraint Graphs w – a weight Label Set = { } – a constraint not OK OK not OK OK not OK The ws sum up to 1. Whole thing is called. val( ) denotes max weight simultaneously satisfiable. X Y Nonlocal Games in foundations of quantum mechanics

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Parallel Repetition: d rounds w – a weight Label Set = { } – a constraint not OK OK not OK OK not OK The ws sum up to 1. Whole thing is called. val( ) denotes max weight simultaneously satisfiable. X Y

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Parallel Repetition: d rounds w – a weight Label Set = { } – a constraint not OK OK not OK OK not OK The ws sum up to 1. Whole thing is called. val( ) denotes max weight simultaneously satisfiable. XdXd YdYd d eg: weight = w 1,14 w 8,20 w 4,13 w 3,17 w 8,18 constraint = 1,14 8,20 4,13 3,17 8,18 d d

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Value under Parallel Repetition val( d ) = val( ) d ? val( d ) · val( ) ? val( 2 ) < val( ) ? val( d ) ! 0 as d ! 1 ? false true false true (took 6 years to prove) True or False?

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Razs Parallel Repetition Theorem Raz 95: val( ) · 1 ) val( d ) · (1 poly( )) d/log(# labels) Tremendously important theorem for proving hardness of approximation results. Holenstein 07: poly( ) can be 3 / Strong Parallel Repetition Problem: can this be improved to ( )?

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The 2-Lin special case # labels = 2, each constraint is either = or Feige-Lovász 91 + Goemans-Williamson 95: val( ) · 1 ) val( d ) · (1 c )) 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.

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

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

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After Parallel Rep: Discrete Torus Graph 5 2 NB: Constraints are unique (x,y) an edge iff ||x-y|| 1 · 1 1 st col. diff., 2 nd col. same 1 st col. diff., 2 nd col. diff. 1 st col. same, 2 nd col. diff. 1 st col. same, 2 nd col. same (self-loops, not pictured) Constraints

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After Parallel Rep: Discrete Torus Graph 5 2 NB: Constraints are unique (x,y) an edge iff ||x-y|| 1 · 1 1 st col. diff., 2 nd col. same 1 st col. diff., 2 nd col. diff. 1 st col. same, 2 nd col. diff. 1 st col. same, 2 nd col. same (self-loops, not pictured) Constraints

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After Parallel Rep: Discrete Torus Graph 5 2 NB: Constraints are unique (x,y) an edge iff ||x-y|| 1 · 1 Given set of Failure Edges, theres a corresp. labeling iff all topologically nontrivial cycles blocked (*)

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

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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:

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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:

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Hence the papers title To understand the truth about parallel repetition, you must get good upper bounds for val( C m 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 d / d Foam Problem.

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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 · m 2 log m, say). Hence val( C m d ) · 1 Proof: A lot of Fourier analysis.

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Our results What we got: val( C m d ) · 1 Best previously: 1 (d) ¢ (1/m) 2 What we really wanted: 1 (d) ¢ (1/m) m = 33

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