A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based @math construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are o (1). Their test makes q *** 3 queries, has amortized query complexity @math , but has an inherent loss of perfect completeness. In this paper we give an (adaptive) hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity @math .
The orthogonal question of designing testers or @math s with as few queries as possible was also considered. In a highly influential paper @cite_8 , H stad constructed a @math system making only three queries. Many variants also followed. In particular @math systems with perfect completeness making three queries were also achieved in @cite_2 @cite_15 . Similar to our approach, O'Donnell and Wu @cite_10 designed an optimal three bit dictatorship test with perfect completeness, and later the same authors constructed a conditional @math system @cite_11 .
@cite_8 @cite_2 @cite_15 @cite_10 @cite_11
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In the conclusion of his monumental paper on optimal inapproximability results, Hastad [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. It is known that there exists a PCP characterization of NP where the verifier makes 3 queries and has a one-sided error that is bounded away from 1; and also that 2 queries do not suffice for such a characterization. Thus PCPs with 3 queries possess non-trivial verification power and motivate the task of determining the lowest error that can be achieved with a 3-query PCP. Recently, Hastad (1997) has shown a tight characterization of NP by constructing a 3-query PCP verifier with "error" arbitrarily close to 1 2. Unfortunately this verifier makes two-sided error and Hastad makes essential use of this feature. One-sided error, on the other hand, is a natural notion to associate with a proof system, since it has the desirable property that every rejected proof has a short counterexample. The question of determining the smallest error for which there exists a 3-query PCP verifier making one-sided error and accepting an NP-complete language, however, remained open. We resolve this question by showing that NP has a 3-query PCP with a one-sided error that is arbitrarily close to 1 2. This characterization is tight, i.e., the error cannot be lower. This result is in seeming contradiction with the results of Trevisan (1997) and Zwick (1998) who show that in order to recognize an NP-complete language, the error probability of a PCP verifier making 3 non-adaptive queries and having one-sided error must be at least 5 8. We get around this bottleneck by designing an adaptive 3-query PCP for NP. Our result yields the first tight analysis of an adaptive PCP; and reveals a previously unsuspected separation between the powers of adaptive and non-adaptive PCPs. Our design and analysis of adaptive PCPs can be extended to higher number of queries as well and we give an example of such a proof system with 5 queries. Our adaptive verifiers yield proof systems whose error probabilities match those of previous constructions, while also achieving one-sidedness in the error. This raises new questions about the power of adaptive PCPs, which deserve further study. We consider the problem of answering queries from databases that may be incomplete. A database is incomplete if some tuples may be missing from some relations, and only a part of each relation is known to be complete. This problem arises in several contexts. For example, systems that provide access to multiple heterogeneous information sources often encounter incomplete sources. The question we address is to determine whether the answer to a specific given query is complete even when the database is incomplete. We present a novel sound and complete algorithm for the answer-completeness problem by relating it to the problem of independence of queries from updates. We also show an important case of the independence problem (and therefore ofthe answer-completeness problem) that can be decided in polynomial time, whereas the best known algorithm for this case is exponential. This case involves updates that are described using a conjunction of comparison predicates. We also describe an algorithm that determines whether the answer to the query is complete in the current state of the database. Finally, we show that our ‘treatment extends naturally to partiallyincorrect databases. Permission to copy without fee all or part of this material is granted provided that the copies aTe not made OT distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, OT to republish, requirea a fee and or special permission from the Endowment. Proceedings of the 22nd VLDB Conference Mumbai(Bombay), India, 1996 We show that a parallel repetition of any two-prover one-round proof system (MIP(2,1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible [U. Feige and O. Verbitsky, Proc.11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 70--76]. We consider the problem of 20 questions with noise for multiple players under the minimum entropy criterion in the setting of stochastic search, with application to target localization. Each player yields a noisy response to a binary query governed by a certain error probability. First, we propose a sequential policy for constructing questions that queries each player in sequence and refines the posterior of the target location. Second, we consider a joint policy that asks all players questions in parallel at each time instant and characterize the structure of the optimal policy for constructing the sequence of questions. This generalizes the single player probabilistic bisection method for stochastic search problems. Third, we prove an equivalence between the two schemes showing that, despite the fact that the sequential scheme has access to a more refined filtration, the joint scheme performs just as well on average. Fourth, we establish convergence rates of the mean-square error and derive error exponents. Finally, we obtain an extension to the case of unknown error probabilities. This framework provides a mathematical model for incorporating a human in the loop for active machine learning systems. In this paper we settle several longstanding open problems in theory of indexability and external orthogonal range searching. In the rst part of the paper, we apply the theory of indexability to the problem of two-dimensional range searching. We show that the special case of 3-sided querying can be solved with constant redundancy and access overhead. From this, we derive indexing schemes for general 4-sided range queries that exhibit an optimal tradeo between redundancy and access overhead. In the second part of the paper, we develop dynamic external memory data structures for the two query types. Our structure for 3-sided queries occupies O(N=B) disk blocks, and it supports insertions and deletions in O(log B N) I Os and queries in O(log B N + T=B) I Os, where B is the disk block size, N is the number of points, and T is the query output size. These bounds are optimal. Our structure for general (4-sided) range searching occupies O (N=B)(log(N=B))= log log B N disk blocks and answers queries in O(log B N + T=B) I Os, which are optimal. It also supports updates in O (log B N)(log(N=B))= log log B N I Os. Center for Geometric Computing, Department of Computer Science, Duke University, Box 90129, Durham, NC 27708 0129. Supported in part by the U.S. Army Research O ce through MURI grant DAAH04 96 1 0013 and by the National Science Foundation through ESS grant EIA 9870734. Part of this work was done while visiting BRICS, Department of Computer Science, University of Aarhus, Denmark. Email: large@cs.duke.edu. yDepartment of Computer Sciences, University of Texas at Austin, Austin, TX 78712-1188. Email vsam@cs.utexas.edu zCenter for Geometric Computing, Department of Computer Science, Duke University, Box 90129, Durham, NC 27708 0129. Supported in part by the U.S. Army Research O ce through MURI grant DAAH04 96 1 0013 and by the National Science Foundation through grants CCR 9522047 and EIA 9870734. Part of this work was done while visiting BRICS, Department of Computer Science, University of Aarhus, Denmark and I.N.R.I.A., Sophia Antipolis, France. Email: jsv@cs.duke.edu. In this paper we present distributed testing algorithms of graph properties in the CONGEST-model [Censor- 2016]. We present one-sided error testing algorithms in the general graph model. We first describe a general procedure for converting @math -testers with a number of rounds @math , where @math denotes the diameter of the graph, to @math rounds, where @math is the number of processors of the network. We then apply this procedure to obtain an optimal tester, in terms of @math , for testing bipartiteness, whose round complexity is @math , which improves over the @math -round algorithm by Censor- (DISC 2016). Moreover, for cycle-freeness, we obtain a of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property. In the second part of the paper we design algorithms for testing whether the network is @math -free for any connected @math of size up to four with round complexity of @math . This improves over the @math -round algorithms for testing triangle freeness by Censor- (DISC 2016) and for testing excluded graphs of size @math by (DISC 2016). In the last part we generalize the global tester by Iwama and Yoshida (ITCS 2014) of testing @math -path freeness to testing the exclusion of any tree of order @math . We then show how to simulate this algorithm in the CONGEST-model in @math rounds. A major obstacle to finding program errors in a real system is knowing what correctness rules the system must obey. These rules are often undocumented or specified in an ad hoc manner. This paper demonstrates techniques that automatically extract such checking information from the source code itself, rather than the programmer, thereby avoiding the need for a priori knowledge of system rules.The cornerstone of our approach is inferring programmer "beliefs" that we then cross-check for contradictions. Beliefs are facts implied by code: a dereference of a pointer, p, implies a belief that p is non-null, a call to "unlock(1)" implies that 1 was locked, etc. For beliefs we know the programmer must hold, such as the pointer dereference above, we immediately flag contradictions as errors. For beliefs that the programmer may hold, we can assume these beliefs hold and use a statistical analysis to rank the resulting errors from most to least likely. For example, a call to "spin_lock" followed once by a call to "spin_unlock" implies that the programmer may have paired these calls by coincidence. If the pairing happens 999 out of 1000 times, though, then it is probably a valid belief and the sole deviation a probable error. The key feature of this approach is that it requires no a priori knowledge of truth: if two beliefs contradict, we know that one is an error without knowing what the correct belief is.Conceptually, our checkers extract beliefs by tailoring rule "templates" to a system --- for example, finding all functions that fit the rule template "a must be paired with b." We have developed six checkers that follow this conceptual framework. They find hundreds of bugs in real systems such as Linux and OpenBSD. From our experience, they give a dramatic reduction in the manual effort needed to check a large system. Compared to our previous work [9], these template checkers find ten to one hundred times more rule instances and derive properties we found impractical to specify manually. Polar codes, as the first provable capacity-achieving error-correcting codes, have received much attention in recent years. However, the decoding performance of polar codes with traditional successive-cancellation (SC) algorithm cannot match that of the low-density parity-check or Turbo codes. Because SC list (SCL) decoding algorithm can significantly improve the error-correcting performance of polar codes, design of SCL decoders is important for polar codes to be deployed in practical applications. However, because the prior latency reduction approaches for SC decoders are not applicable for SCL decoders, these list decoders suffer from the long-latency bottleneck. In this paper, we propose a multibit-decision approach that can significantly reduce latency of SCL decoders. First, we present a reformulated SCL algorithm that can perform intermediate decoding of 2 b together. The proposed approach, referred as 2-bit reformulated SCL ( 2b-rSCL ) algorithm , can reduce the latency of SCL decoder from ( @math ) to ( @math ) clock cycles without any performance loss. Then, we extend the idea of 2-b-decision to general case, and propose a general decoding scheme that can perform intermediate decoding of any @math bits simultaneously. This general approach, referred as @math -bit reformulated SCL ( @math b-rSCL ) algorithm , can reduce the overall decoding latency to as short as @math cycles. Furthermore, on the basis of the proposed algorithms, very large-scale integration architectures for 2b-rSCL and 4b-rSCL decoders are synthesized. Compared with a prior SCL decoder, the proposed (1024, 512) 2b-rSCL and 4b-rSCL decoders can achieve 21 and 60 reduction in latency, 1.66 and 2.77 times increase in coded throughput with list size 2, and 2.11 and 3.23 times increase in coded throughput with list size 4, respectively. Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal. We consider the problem of single-round private information retrieval (PIR) from @math replicated databases. We consider the case when @math databases are outdated (unsynchronized), or even worse, adversarial (Byzantine), and therefore, can return incorrect answers. In the PIR problem with Byzantine databases (BPIR), a user wishes to retrieve a specific message from a set of @math messages with zero-error, irrespective of the actions performed by the Byzantine databases. We consider the @math -privacy constraint in this paper, where any @math databases can collude, and exchange the queries submitted by the user. We derive the information-theoretic capacity of this problem, which is the maximum number of that can be retrieved privately (under the @math -privacy constraint) for every symbol of the downloaded data. We determine the exact BPIR capacity to be @math , if @math . This capacity expression shows that the effect of Byzantine databases on the retrieval rate is equivalent to removing @math databases from the system, with a penalty factor of @math , which signifies that even though the number of databases needed for PIR is effectively @math , the user still needs to access the entire @math databases. The result shows that for the unsynchronized PIR problem, if the user does not have any knowledge about the fraction of the messages that are mis-synchronized, the single-round capacity is the same as the BPIR capacity. Our achievable scheme extends the optimal achievable scheme for the robust PIR (RPIR) problem to correct the introduced by the Byzantine databases as opposed to in the RPIR problem. Our converse proof uses the idea of the cut-set bound in the network coding problem against adversarial nodes. Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz. Goldreich (ECCC 2000) suggested a simple construction of a candidate one-way function f : 0, 1 n → 0, 1 m where each bit of output is a fixed predicate P of a constant number d of (random) input bits. We investigate the security of this construction in the regime m = Dn, where D(d) is a sufficiently large constant. We prove that for any predicate P that correlates with either one or two of its inputs, f can be inverted with high probability. We also prove an amplification claim regarding Goldreich’s construction. Suppose we are given an assignment @math that has correlation @math with the hidden assignment @math . Then, given access to x′, it is possible to invert f on x with high probability, provided @math is sufficiently large. Recently an orthogonal basis ofWN -algebra (AFLT basis) labeled by N-tuple Young diagrams was found in the context of 4D 2D duality. Recursion relations among the basis are summarized in the form of an algebra SH c which is universal for any N. We show that it has an S3 automorphism which is referred to as triality. We study the level- rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by N or M Young diagrams with the rows of boxes appropriately shued. The reshuing of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH c . Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the N-Burge condition in the Hilbert space. 1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 We study the problem of Private Information Retrieval (PIR) in the presence of prior side information. The problem setup includes a database of @math independent messages possibly replicated on several servers, and a user that needs to retrieve one of these messages. In addition, the user has some prior side information in the form of a subset of @math messages, not containing the desired message and unknown to the servers. This problem is motivated by practical settings in which the user can obtain side information opportunistically from other users or has previously downloaded some messages using classical PIR schemes. The objective of the user is to retrieve the required message without revealing its identity while minimizing the amount of data downloaded from the servers. We focus on achieving information-theoretic privacy in two scenarios: (i) the user wants to protect jointly its demand and side information; (ii) the user wants to protect only the information about its demand, but not the side information. To highlight the role of side information, we focus first on the case of a single server (single database). In the first scenario, we prove that the minimum download cost is @math messages, and in the second scenario it is @math messages, which should be compared to @math messages, the minimum download cost in the case of no side information. Then, we extend some of our results to the case of the database replicated on multiple servers. Our proof techniques relate PIR with side information to the index coding problem. We leverage this connection to prove converse results, as well as to design achievability schemes. We consider the makespan-minimization problem on unrelated machines in the context of algorithmic mechanism design. No truthful mechanisms with non-trivial approximation guarantees are known for this multidimensional domain. We study a well-motivated special case (also a multidimensional domain), where the processing time of a job on each machine is either "low" or "high." We give a general technique to convert any c-approximation algorithm (in a black-box fashion) to a 3c-approximation truthful-in-expectation mechanism. Our construction uses fractional truthful mechanisms as a building block, and builds upon a technique of Lavi and Swamy [Lavi, R., Swamy, C., 2005. Truthful and near-optimal mechanism design via linear programming. In: Proc. 46th FOCS, pp. 595-604]. When all jobs have identical low and high values, we devise a deterministic 2-approximation truthful mechanism. The chief novelty of our results is that we do not utilize explicit price definitions to prove truthfulness. Instead we design algorithms that satisfy cycle monotonicity [Rochet, J., 1987. A necessary and sufficient condition for rationalizability in a quasilinear context. J. Math. Econ. 16, 191-200], a necessary and sufficient condition for truthfulness in multidimensional settings; this is the first work that leverages this characterization. We consider the performance of two algorithms, GUC and SC studied by M. T. Chao and J. Franco SIAM J. Comput.15(1986), 1106?1118;Inform. Sci.51(1990), 289?314 and V. Chvatal and B. Reed in“Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, 1992,” pp. 620?627, when applied to a random instance ? of a boolean formula in conjunctive normal form withnvariables and ?cn? clauses of sizekeach. For the case wherek=3, we obtain the exact limiting probability that GUC succeeds. We also consider the situation when GUC is allowed to have limited backtracking, and we improve an existing threshold forcbelow which almost all ? is satisfiable. Fork?4, we obtain a similar result regarding SC with limited backtracking.