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3-Query Dictator Testing

The joint work with Carnegie Mellon University explores the computational complexity and approximability of Max-3CSP Dictator Testing problems. It delves into the difficulty levels of satisfying optimal solutions and examines algorithms for solving instances. The research discusses open problems and conjectures regarding the applicability of Fourier transforms in analyzing Dictator Tests.

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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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