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3-Query Dictator Testing

3-Query Dictator Testing. Ryan O’Donnell. Yi Wu. joint work with. Carnegie Mellon University. Carnegie Mellon University. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A. Motivation: Max-3CSP. Constraint Satisfaction Problems (CSPs). Input:.

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3-Query Dictator Testing

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  1. 3-Query Dictator Testing Ryan O’Donnell Yi Wu joint work with Carnegie Mellon University Carnegie Mellon University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

  2. Motivation: Max-3CSP

  3. Constraint Satisfaction Problems (CSPs) Input: Output: Assignment: vi2 {0,1} Desideratum: Satisfy as much as possible. w1 w2 Definition: 0 · OPT · 1 is max. possible w3 Definition: · k vbls per constraint: = “Max-kCSP” w4 w5 w6 Fixing “type” of constraints  special cases: w7 w8 Max-3Sat Max-3Lin w9 + ¢ ¢ ¢ ¢¢¢ ¢ ¢ ¢ ¢ ¢ ¢ = 1

  4. Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm Other CSPs (essentially) Max-2SatMax-3SatMax-kSatMax-kLinMax-kCSPMax-CutMax-Directed-CutMin-BisectionSparsest-CutBalanced-SeparatorVertex-CoverIndependent-SetCliqueApproximate-Graph-ColoringMin-Multiway-CutMetric-Labeling0-ExtensionCut-Norm

  5. Max-3CSP Input: Output: Assignment: vi2 {0,1} Desideratum: Satisfy as much as possible. w1 w2 Definition: 0 · OPT · 1 is max. possible w3 Definition: · 3 vbls per constraint: = “Max-3CSP” w4 w5 w6 w7 w8 w9 + ¢ ¢ ¢ ¢¢¢ = 1

  6. Computational Complexity of CSPs 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.

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

  8. Open Problems [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 NTWclauses.” [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. NTW(a,b,c) = 1 , # 1’s among a,b,c is zero, one, or three – i.e., Not Two “ ”

  9. Dictator Testing(AKA Long Code testing)

  10. Dictator Testing [BGS95] Property Testing problem Query access to unknown Boolean function f : {0,1}n {0,1} Want to test if f is a Dictator: f(x1, …, xn) = xifor 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

  11. 3-Query Dictator Testing x, y, z Tester randomly chooses: i) 3 strings,x, y, z2 {0,1}n, ii) a 3-bit predicate, φ:{0,1}3→ {acc, rej} f : {0,1}n {0,1} f(x), f(y), f(z) “accepts” iff φ(f(x), f(y), f(z)) = acc “Tester uses predicate set Φ”$ Φ = {possible φ’s tester may choose} “Completeness” ¸ c$ all n Dictators accepted w. prob. ¸ c “Soundness” · s$ “very non-Dictatorial f” accepted “w. prob. · s + o(1)”

  12. 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/±.

  13. 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)

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

  15. Connection to Inapproximability 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.)

  16. Implication for Max-3CSP [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 NTWclauses.” [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. “ ”

  17. Our Results 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”…??

  18. Our NTW-based test: how and why

  19. 3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW x ) f ( NTW( y f ( ) z f ( ) ) D = = p q r s t w. prob. Test: Choose triple (x, y, z) from D­n. Problem: Constant functions Solution: By “odd-izing” (“folding”) trick, may assume f(:x) = :f(x) Issue: Reqs. uniform distr. on x, y, z

  20. 3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW x ) f ( NTW( y f ( ) z f ( ) ) D = = p w. prob. Test: Choose triple (x, y, z) from D­n. Problem: Majority Corr[xi, yi] = Pr[xi = yi] – Pr[xiyi] = 2p Solution: Make p very small

  21. 3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW x ) f ( NTW( y f ( ) z f ( ) ) D = = w. prob. Test: Choose triple (x, y, z) from D­n. Problem? Large odd Parity Solution: Don’t take ± = 0!

  22. 3-Query Dictator-vs.-quasirandom Testing Upper Bound using NTW x ) f ( NTW( y f ( ) z f ( ) ) D = = ± w. prob. Test: Choose triple (x, y, z) from D±­n. Fact: D = (1 – ±) D + ±D ± EQU XOR Equivalent test: 1. Form “random restriction” fw with ¤-probability 1 – ±. 2. Do BLR test on fw, but also accept (0,0,0).

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

  24. Open Problems

  25. Open Problems 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?

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