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3-Query Dictator Testing Ryan O’Donnell Carnegie Mellon University joint work with Yi Wu TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A Carnegie Mellon University

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Motivation: Max-3CSP

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Constraint Satisfaction Problems (CSPs) Input: ¢ ¢ ¢ Output: Assignment: v i 2 {0,1} Desideratum: Satisfy as much as possible. w1w1 w2w2 w3w3 w4w4 w5w5 w6w6 w7w7 w8w8 w9w9 ¢¢¢ + = 1 Definition: 0 · OPT · 1 is max. possible Definition: · k vbls per constraint: = “Max-kCSP” Fixing “type” of constraints special cases: Max-3Sat Max-3Lin ¢ ¢ ¢

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Max-2Sat Max-3Sat Max-kSat Max-kLin Max-kCSP Max-Cut Max-Directed-Cut Min-Bisection Sparsest-Cut Balanced-Separator Vertex-Cover Independent-Set Clique Approximate-Graph-Coloring Min-Multiway-Cut Metric-Labeling 0-Extension Cut-Norm Other CSPs (essentially)

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Max-3CSP Input: ¢ ¢ ¢ Output: Assignment: v i 2 {0,1} Desideratum: Satisfy as much as possible. w1w1 w2w2 w3w3 w4w4 w5w5 w6w6 w7w7 w8w8 w9w9 ¢¢¢ + = 1 Definition: 0 · OPT · 1 is max. possible Definition: · 3 vbls per constraint: = “Max-3CSP”

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Max-Blah is c vs. s easy: satisfying ¸ s when OPT ¸ c is in poly time. Max-Blah is c vs. s hard: satisfying ¸ s when OPT ¸ c is NP-hard. Computational Complexity of CSPs

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Approximability of Max-3CSP 1 s c 0 1 (OPT) [Cook71] = NP-hard [Johnson74] 1/8 = in poly time [AS, ALMSS92] [BGS95] (.96) [Trevisan96] 1/4 [TSSW96] (.367) [Håstad97] 3/4 [Trevisan97] (.514) [Zwick98,02] 1/2 5/8 [KS06] (.74)

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[Zwick98], on his 1 vs. 5/8 easiness result for Max-3CSP: “ We conjecture that this result is optimal. ” “ … the hardest satisfiable instances of Max-3CSP [for the algorithm] turn out to be instances in which all clauses are NTW clauses. ” [Håstad97], p. 65, Concluding remarks: The technique of using Fourier transforms to analyze [Dictator Tests] seems very strong. It does not, however, seem universal even limited to CSPs. In particular, an open question that remains is to decide whether the NTW predicate is non-approximable beyond the random assignment threshold [5/8] on satisfiable instances. Open Problems NTW (a,b,c) = 1, # 1’s among a,b,c is zero, one, or three – i.e., Not Two ” “

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Dictator Testing (AKA Long Code testing)

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Property Testing problem Query access to unknown Boolean function f : {0,1} n {0,1} Want to test if f is a Dictator: f(x 1, …, x n ) = x i for some i. Can only make a constant number of queries And by constant, I mean 3 Or fewer And the queries must be non-adaptive Dictator Testing[BGS95]

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3-Query Dictator Testing randomly chooses: i) 3 strings, x, y, z 2 {0,1} n, ii) a 3-bit predicate, φ :{0,1} 3 → {acc, rej} x, y, z f(x), f(y), f(z) “accepts” iff φ(f(x), f(y), f(z)) = acc “Completeness” ¸ c $ all n Dictators accepted w. prob. ¸ c “Soundness” · s $ “very non-Dictatorial f” accepted “w. prob. · s + o(1)” Tester “Tester uses predicate set Φ” $ Φ = {possible φ’s tester may choose}

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Soundness Condition Usually: “Every f which is ± -far from all Dictators is accepted w. prob. · s.” [Håstad97]: Too hard! Relax. Definition: f is quasirandom if fixing any O(1) input bits changes bias by at most o(1). Remark: Dictators are the epitome of not being quasirandom. Formally: f is ( ², ± )-quasirandom if for all 0 < |S| · 1/ ±.

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Quasirandomness Definition: f is quasirandom if fixing any O(1) input bits changes bias by at most o(1). Not quasirandom:Dictators “Juntas” Epitome of quasirandom: Constants (f ´ 0, f ´ 1) Majority Large Parities: f(x) = where |S| > ω(1)

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Dictator-vs.-quasirandom Tests “Dictator-vs.-quasirandom” Tests: Formally: Given a sequence of tests ( T n ), Soundness · s $ every quasirandom f accepted w. prob. · s + o(1) Soundness · s $ for all ´ > 0, exists ², ± > 0, for all suff. large n, T n accepts every ( ², ± )-quasirandom f w. prob. · s + ´

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Meta-Theorem: Suppose you build a Dictator-vs.-quasirandom test with: completeness ¸ c, soundness · s, tester uses predicate set Φ. Then Max-Φ is c vs. s + ² hard. (Max–Φ is the CSP where all constraints are from the set Φ.) Connection to Inapproximability

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[Zwick98], on his 1 vs. 5/8 easiness result for Max-3CSP: “ We conjecture that this result is optimal. ” “ … the hardest satisfiable instances of Max-3CSP [for the algorithm] turn out to be instances in which all clauses are NTW clauses. ” [Håstad97], p. 65, Concluding remarks: The technique of using Fourier transforms to analyze [Dictator Tests] seems very strong. It does not, however, seem universal even limited to CSPs. In particular, an open question that remains is to decide whether the NTW predicate is non-approximable beyond the random assignment threshold [5/8] on satisfiable instances. Implication for Max-3CSP ” “

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Theorem: a. There is a 3-query Dictator-vs.-quasirandom test, using NTW predicate, with completeness c = 1 and soundness s = 5/8. [Pf: Fourier analysis.] b. Every 3-query Dictator-vs.-quasirandom test, using any mix of predicates, with completeness c = 1 has soundness s ¸ 5/8. [Pf: Uses Zwick’s SDP alg.] Not a Theorem: Max- NTW is 1 vs. 5/8 hard. Why? Meta-Theorem problematic… maybe with Khot’s “2-to-1 Conjecture”…?? Our Results

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Our NTW -based test: how and why

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3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW f ( NTW ( 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 pqrst Test: Choose triple (x, y, z) from D n. D = w. prob. xixi yiyi zizi = Solution: By “odd-izing” (“folding”) trick, may assume f( : x) = : f(x) Issue: Reqs. uniform distr. on x, y, z )) ) ) z y x

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3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW f ( NTW ( 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 p Test: Choose triple (x, y, z) from D n. D = w. prob. xixi yiyi zizi = Corr[x i, y i ] = Pr[x i = y i ] – Pr[x i y i ] = 2p Solution: Make p very small )) ) ) z y x

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3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW f ( NTW ( 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 Test: Choose triple (x, y, z) from D n. D = w. prob. xixi yiyi zizi = Solution: Don’t take ± = 0! )) ) ) z y x

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3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW f ( NTW ( 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 Test: Choose triple (x, y, z) from D ± n. D = w. prob. xixi yiyi zizi = )) ) ) z y x ± D = ± Fact: (1 – ± ) D + ± D XOR EQU Equivalent test: 1. Form “random restriction” f w with ¤ -probability 1 – ±. 2. Do BLR test on f w, but also accept (0,0,0).

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Analyzing the Test Pr[acc. odd f] · Håstad’s term: · ± when f is ( ± 2, ± 2 )-quasirandom Handle with careful use of the “hypercontractive inequality” Long story short: last term always

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Open Problems

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Prove Max-3CSP is 1 vs. 5/8 + ² hard. Prove Max-3CSP is 1 vs. 5/8 + ² hard assuming Khot’s 2-to-1 Conjecture. Tackle Max-2Sat. [cf. Austrin07a, Austrin07b] Max-4CSP? Open Problems

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