3 - Bit Dictator Testing : 1 vs . 5 / 8 Ryan O ’ Donnell

In the conclusion of his monumental paper on optimal inapproximability results, H̊astad [13] suggested that Fourier analysis of Dictator (Long Code) Tests may not be universally applicable in the study of CSPs. His main open question was to determine if the technique could resolve the approximability of satisfiable 3-bit constraint satisfaction problems. In particular, he asked if the “Not Two” (NTW) predicate is non-approximable beyond the random assignment threshold of 5/8 on satisfiable instances. Around the same time, Zwick [30] showed that all satisfiable 3-CSPs are 5/8-approximable and conjectured that the 5/8 is optimal. In this work we show that Fourier analysis techniques can produce a Dictator Test based on NTW with completeness 1 and soundness 5/8. Our test’s analysis uses the Bonami-Gross-Beckner hypercontractive inequality. We also show a soundness lower bound of 5/8 for all 3-query Dictator Tests with perfect completeness. This lower bound for Property Testing is proved in part via a semidefinite programming algorithm of Zwick [30]. Our work precisely determines the 3-query “Dictatorship Testing gap”. Although this represents progress on Zwick’s conjecture, current PCP “outer verifier” technology is insufficient to convert our Dictator Test into an NP-hardness-of-approximation result. 1 Dictator Testing, and its motivation In this paper we study a Property Testing problem called Dictator Testing. Dictator Testing is strongly motivated by its applications to proving NP-hardnessof-approximation results (in which context it is often called “Long Code Testing”). We describe the Dictator Testing problem informally in this section; for formal definitions, see Section 4.1. In Dictator Testing, we have black-box query access to an unknown boolean function f : {0, 1} → {0, 1}. The goal is to test the extent to which f is close to a “dictator” function; i.e., one of the n functions of the form f(x1, . . . , xn) = xi. ∗Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213. Supported in part by NSF CAREER grant CCF-0747250 and a CyLab Seed Grant. Recall that a “test” is a randomized algorithm which makes a very small number of queries to f and then either “accepts” or “rejects”. It is “nonadaptive” if it determines all query strings before seeing the responses. The Dictator Testing problem was first studied by Bellare, Goldreich, and Sudan [1], with hardness-of-approximation as the motivation. It was later independently introduced, with the “dictator” terminology, by Parnas, Ron, and Samorodnitsky [24]. Definition 1.1. A Dictator Test has completeness c if all n dictator functions are accepted with probability at least c. We say a Dictator Test has perfect completeness if it has completeness 1. In this paper we consider only nonadaptive Dictator Tests with perfect completeness, unless otherwise specified. As for the “soundness” of a Dictator Test, we briefly discuss several possible criteria one could require: Local testability: In this model of soundness, any function f which is -far from every dictator should be accepted with probability at most 1 − Ω( ). Here we hope to make a very small constant number of queries, such as 3 or 4. Bellare, Goldreich, and Sudan [1] (BGS) implicitly gave a 4-query (indeed, 3-query adaptive) test; for a simple 3-query construction, see e.g. [22]. Such “Local Dictator Tests” play a useful role in Dinur’s proof of the PCP Theorem [4]. Usual Property Testing soundness: In this model, the tester is also given a parameter ; any function f which is -far from every dictator should be accepted with probability at most, say, 1/3. Here we hope to make few queries as a function of . Note that by repeating a local test O(1/ ) times, we get an O(1/ )query Dictator Test with the usual Property Testing soundness. Rejecting very far functions: In this model, soundness is only concerned with functions f which are (1/2 − o(1))-far from every dictator; i.e., have correlation at most o(1) with every dictator. The goal is to accept such functions with as low a probability as possible, while making a very small constant number of queries. For example, the 4-query BGS test accepts these functions with probability at most 17/20 + o(1). The major contribution of BGS was showing that Dictator Tests with this kind of soundness can yield strong NP365 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. hardness-of-approximation results for constraint satisfaction problems (“CSPs”). Rejecting “quasirandom” functions: H̊astad [12, 13] introduced this relaxation of the above. One can think of it as only requiring soundness for functions f which have correlation at most o(1) with every “junta” (function depending on only O(1) coordinates). We refer to such f ’s as “quasirandom”, and such tests as “Dictator-vs.-Quasirandom Tests”. Definition 1.2. (Informal.) A Dictator-vs.Quasirandom Test has soundness at most s if every quasirandom function is accepted with probability at most s+ o(1). For example, H̊astad [13] gave a (nonadaptive) 3-query Dictator-vs.-Quasirandom Test with soundness 3/4. As H̊astad and others have demonstrated, Dictator-vs.-Quasirandom Tests can often be used to prove optimal inapproximability results for CSPs. 1.1 Optimal approximability for k-CSPs. The major motivation for Dictator Testing is proving hardness-of-approximation results for CSPs. We discuss this connection in Section 1.2; however first let us describe k-CSPs. A (boolean) “k-CSP” is a system of constraints over n boolean-valued variables vi in which each constraint involves at most k of the variables. We also assume each constraint has a nonnegative weight, with the sum of all weights being 1. Given a k-CSP, the natural algorithmic task, called “Max-kCSP”, is to find an assignment to the variables such that the total weight of satisfied constraints is as large as possible. We write “Opt” to denote the weight satisfied by the best possible assignment. We also say that a CSP is “satisfiable” if Opt = 1. Our main motivation in this paper is studying the difficulty of satisfiable Max-3CSP instances, of which the following is a small example: weight: constraint: 1/4 v1 ∧ ¬v3 ∧ v4 1/4 IF v3 THEN v4 ELSE ¬v5 1/2 v2 6= v5 Each constraint in a k-CSP is of a certain “type”; more precisely, it is a certain predicate of arity at most k over the variables. If we specialize Max-kCSP by restricting the type of constraints allowed, we get some of the most canonical NP optimization problems. For example: • Max-2Sat: only the four predicates vi∨vj , vi∨¬vj , ¬vi ∨ vj , ¬vi ∨ ¬vj ; • Max-3Lin: only the two predicates vi ⊕ vj ⊕ vk, ¬(vi ⊕ vj ⊕ vk); • Max-Cut: only the predicate vi 6= vj . If we restrict the allowed predicates to some set Φ, we call the associated problem Max-Φ. Determining Opt for these CSP families is NP-hard, but there is an enormous literature on polynomial-time approximation algorithms. To complement this, we can also look for NP-hardness-of-approximation results. As we describe in the next section, all of the best known inapproximability results rely critically on Dictator Testing. We now have optimal (i.e., matching) approximation algorithms and NP-hardness-of-approximation results for some key problems: Max-kLin(mod q) for k ≥ 3 [13], Max-3Sat [13, 15, 31], and a few other MaxkCSP problems with k ≥ 3 [13, 30, 29, 9]. However, many basic problems remain unresolved; for example, we do not know if 90%-approximating Max-Cut is in P or is NP-hard. Similarly, given a satisfiable 3-CSP, we do not know if satisfying 2/3 of the constraint-weight is in P or is NP-hard. 1.2 Dictator Testing and inapproximability. There is a close connection between CSPs and the Property Testing of boolean functions. To illustrate this, suppose T is a nonadaptive 3-query Dictator-vs.Quasirandom Test on functions f : {0, 1} → {0, 1}. Imagine we consider all possible random bit choices of T , and in each case write down the (up to) 3 strings x, y, z queried and the predicate applied to the outcomes to decide accept/reject. The complete behavior of T might then look like the following: w.p. p1, accept iff f(x) ∨ f(y) ∨ f(z), w.p. p2, accept iff ¬f(x) ∨ f(y), w.p. p3, accept iff ¬f(x) ∨ ¬f(y) ∨ ¬f(z), etc. This is precisely an instance of Max-3CSP, in which the “variables” are the f(x)’s. Note that the weights pi indeed sum up to 1. More generally, if T makes at most q nonadaptive queries it can be viewed as an instance of Max-qCSP. Further, suppose that T “uses the predicate set Φ” — i.e., its decision to accept/reject is always based on applying a predicate from the set Φ to its query responses. Then T can be viewed as an instance of Max-Φ. The above example illustrates a tester which uses the set of ORs on up to 3 literals; thus it can be viewed as an instance of Max-3Sat. Bellare, Goldreich, and Sudan [1] and H̊astad [12, 13] showed how this connection can be used to create “gadgets” for NP-hardness-ofapproximation reults. Their work led to the following well-known “Rule of Thumb”: Rule of Thumb. For the Max-Φ problem, to prove that distinguishing Opt ≥ c and Opt ≤ s + 366 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. is NP-hard, construct a nonadaptive Dictator-vs.Quasirandom Test using Φ, with completeness c and soundness s. For example, the key step in H̊astad’s famous (7/8+ )-hardness result for satisfiable Max-3Sat instances was his construction of a 3-query Dictator-vs.-Quasirandom Test using OR tests, with perfect completeness and soundness 7/8. Actual theorems based on the Rule of Thumb are recalled in the full version of this paper. 2 Our contribution: optimal 3-query tests with perfect completeness 2.1 Satisfiable 3-CSPs. One of the most notable open questions in the area of CSP approximability is that of analyzing satisfiable 3-CSPs: Question 1: Given a satisfiable 3-CSP, can we efficiently satisfy constraint-weight at least s? Zwick [30] made a comprehensive study of the Max-3CSP problem and gave an efficient algorithm which 5/8-satisfies any satisfiable 3-CSP instance. (This improved and built upon the earlier .514-algorithm of Trevisan [28].) Zwick conjectured that this algorithm is optimal; i.e., obtaining 5/8 + is NP-hard for all constant > 0. In the language of Probabilistically Checkable Proofs, Zwick’s conjecture states that NP ⊆ naPCP1,5/8+ (O(log n), 3). H̊astad’s contemporaneous treatise on optimal inapproximability [11] gave an NP-hardness result for s > 3/4, and improving this was left as the main open problem in his work. Almost a decade later, no progress had been made on closing the gap, and H̊astad reposed the problem [14]. Shortly thereafter, Khot and Saket [19] achieved the first improved hardness, showing that satisfiable 3-CSP instances are NP-hard to approximate to any factor better than 20/27 ≈ .74. In originally posing the problem, H̊astad suggested that Fourier analysis of Dictator Tests does not seem universally applicable in the study of CSPs. The associated Dictator-vs.-Quasirandom Property Testing question here is particularly easy to state: Question 2: What is the least possible soundness of a nonadaptive 3-query Dictator-vs.-Quasirandom Test with perfect completeness? Khot and Saket’s result yields an upper bound of 20/27 for this question; the question of proving a lower bound has not been explicitly considered. The main result in this paper is an exact answer to Question 2: Theorem 2.1. (Main results, informally stated.) 1. There is a nonadaptive 3-query Dictator-vs.Quasirandom Test with perfect completeness and soundness 5/8. The test uses only the “Not-Two” (NTW) predicate. 2. Every nonadaptive 3-query Dictator-vs.Quasirandom Test with perfect completeness has soundness at least 5/8. The upper bound in Theorem 2.1 is proved in Section 6; Fourier analysis is indeed the key to the proof. Due to space limitations, the lower bound in Theorem 2.1 is deferred to the full version of this paper. The NTW predicate. NTW is the 3-bit predicate which is satisfied if the number of True inputs is either zero, one, or three — i.e., not two. Our test actually uses all eight NTW predicates gotten by allowing the query responses to be negated. Although Zwick’s algorithm satisfies 5/8 of the constraints in any 3-CSP, even with mixed “types” of constraints, the bottleneck predicate for him is the NTW constraint. H̊astad’s open question more specifically asked whether or not the NTW predicate is satisfiable beyond the random-assignment threshold of 5/8 on satisfiable instances. What we don’t prove. Unfortunately, the formal theorems behind the Rule of Thumb are not sufficient to convert our Dictator Test into a 5/8 + NP-hardness result for satisfiable Max-NTW instances, and thus prove Zwick’s conjecture. See the discussions in Section 7 and the full version of this paper. However in an upcoming work building on the present paper, we will show that this result can be obtained assuming Khot’s “d-to-1 Conjecture” ([16]) for any constant d. 2.2 Methods. Upper bound. Given the task of constructing an NTW-based Dictator-vs.-Quasirandom Test with perfect completeness, we describe in Appendix ?? how the correct test is almost “forced” upon us. Thus the main task is in the analysis of this test. For this we use some slightly tricky Fourier analysis, including the hypercontractive inequality [3]. This is the first Dictator Testing result we are aware of that uses the hypercontractive inequality without using the Invariance Principle [21]. Indeed, the Invariance Principle does not seem useful for our result, and the fact that we use the hypercontractive inequality directly gives us an exponentially better tradeoff between soundness and influences than those given by Invariance-based analyses. 367 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. Lower bound. The problem of proving lower bounds for Dictator Tests (of the sort needed for inapproximability) does not seem to have been considered until extremely recently. In [23], the present authors observed that because of the “Rule of Thumb” described in Section 1.2, the existence of strong approximation algorithms “ought to” imply Dictator-vs.-Quasirandom Testing lower bounds. We discuss this connection in the context of related work in Section 3. We show this directly for our problem, as was done in a much simpler context in [23]. Our proof of the soundness lower bound in Theorem 2.1 involves using Zwick’s algorithm to show the existence a quasirandom function passing any given perfect-completeness Dictator Test with probability at least 5/8− o(1); this quasirandom function is either a random threshold function or a random odd parity. Zwick’s algorithm in part uses semidefinite programming, and we feel that the use of semidefinite programming in Property Testing lower bounds is an interesting method.