We describe a new parameterized family of symmetric error-correcting codes with low-density parity-check matrices (LDPC). Our codes can be described in two seemingly different ways. First, in relation to Reed-Muller codes: our codes are functions on a subset of the points in 𝔽ⁿ whose restrictions to a prescribed set of affine lines has low degree. Alternatively, they are Tanner codes on high dimensional expanders, where the coordinates of the codeword correspond to triangles of a 2-dimensional expander, such that around every edge the local view forms a Reed-Solomon codeword. For some range of parameters our codes are provably locally testable, and their dimension is some fixed power of the block length. For another range of parameters our codes have distance and dimension that are both linear in the block length, but we do not know if they are locally testable. The codes also have the multiplication property: the coordinate-wise product of two codewords is a codeword in a related code. The definition of the codes relies on the construction of a specific family of simplicial complexes which is a slight variant on the coset complexes of Kaufman and Oppenheim. We show a novel way to embed the triangles of these complexes into 𝔽ⁿ, with the property that links of edges embed as affine lines in 𝔽ⁿ. We rely on this embedding to lower bound the rate of these codes in a way that avoids constraint-counting and thereby achieves non-trivial rate even when the local codes themselves have arbitrarily small rate, and in particular below 1/2.

Abstract. Agreement tests are a generalization of low degree tests that capture a local-to-global phenomenon, which forms the combinatorial backbone of most probabilistically checkable proof (PCP) constructions. In an agreement test, a function is given by an ensemble of local restrictions. The agreement test checks that the restrictions agree when they overlap, and the main question is whether average agreement of the local pieces implies that there exists a global function that agrees with most local restrictions. There are very few structures that support agreement tests, essentially either coming from algebraic low degree tests or from direct product tests (and recently also from high-dimensional expanders). In this work, we prove a new agreement theorem which extends direct product tests to higher dimensions, analogously to how low degree tests extend linearity testing. As a corollary of our main theorem, it follows that an ensemble of small graphs on overlapping sets of vertices can be glued together to one global graph assuming they agree with each other on average. We prove the agreement theorem by (re)proving the agreement theorem for dimension 1, and then generalizing it to higher dimensions (with the dimension 1 case being the direct product test and dimension 2 being the graph case). A key technical step in our proof is the reverse union bound, which allows us to treat dependent events as if they are disjoint, and may be of independent interest. An added benefit of the reverse union bound is that it can be used to show that the “majority decoded” function also serves as a global function that explains the local consistency of the agreement theorem, a fact that was not known even in the direct product setting (dimension 1) prior to our work. Beyond the motivation to understand fundamental local-to-global structures, our main theorem allows us to lift structure theorems from the standard uniform distribution [Formula: see text] to the [Formula: see text]-biased distribution [Formula: see text]. As a simple demonstration of this paradigm, we show how the low degree testing result of Alon et al. [ IEEE Trans. Inform. Theory, 51 (2005), pp. 4032–4039,] and Bhattacharyya et al. [ Proc. 51 st FOCS, IEEE, 2010, pp. 488–497], originally proved for [Formula: see text], can be extended to the [Formula: see text]-biased hypercube [Formula: see text], even for very small subconstant [Formula: see text].

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.

We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work. Derandomized direct product testing, also known as agreement testing, is the following problem. Let$X$be a family of k-element subsets of$[N]$and let$\{f_{s}:s\rightarrow\Sigma\vert s\in X\}$be an ensemble of local functions, each defined over a subset$s\subset\lceil N$. Suppose that we run the following so-called agreement test: choose a random pair of sets$s_{1}, s_{2}\in X$that intersect on$\sqrt{k}$elements, and accept if$f_{s_{1}}, f_{s_{2}}$agree on the elements in$s_{1}\cap s_{2}$. We denote the success probability of this test by Agree$\{f_{s}\})$Given that Agree$(\{f_{s}\})=\varepsilon > 0$is there a global function$G:[N]\rightarrow\Sigma$such that$f_{s}=G\vert _{s}$for a non-negligible fraction of$s\in X\ ?$We construct a family$X$of k-subsets of$[N]$such that$\vert X\vert =O(N)$, and such that it satisfies the low acceptance agreement theorem. Namely,$\text{Agree}\left(\left\{f_s\right\}\right)>\varepsilon \Longrightarrow \exists G:[N] \rightarrow \Sigma, \quad \underset{s}{\mathbb{P}}\left[\left.f_s \stackrel{0.99}{\approx} G\right\vert_s\right] \geqslant \text{poly}(\varepsilon)$. A key idea is to replace the well-studied LSV complexes by symplectic high dimensional expanders (HDXs). The family$X$is just the k-faces of the new symplectic HDXs. The latter serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers. We also give a polynomial-time algorithm to construct this family of sym-plectic HDXs.

We introduce a high-dimensional cubical complex, for any dimension$t \in \mathbb{N}$, and apply it to the design of quantum locally testable codes. Our complex is a natural generalization of the constructions by Panteleev and Kalachev and by Dinur et. al of a square complex (case$t=2$), which have been applied to the design of classical locally testable codes (LTC) and quantum low-density parity check codes (qLDPC) respectively. We turn the geometric (cubical) complex into a chain complex by relying on constant-sized local codes$h_{1}, \ldots,h_{t}$as gadgets. A recent result of Panteleev and Kalachev on existence of tuples of codes that are product expanding enables us to prove lower bounds on the cycle and co-cycle expansion of our chain complex. For$t=4$our construction gives a new family of “almost-good” quantum LTCs - with constant relative rate, inverse-polylogarithmic relative distance and soundness, and constant-size parity checks. Both the distance of the quantum code and its local testability are proven directly from the cycle and co-cycle expansion of our chain complex.

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. We also study the closely related swap coboundary expansion. Swap cosystolic expansion is defined by considering, for a given complex X, its faces complex , whose vertices are r-faces of X and where two vertices are connected if their disjoint union is also a face in X. The faces complex is a derandomization of the product of X with itself r times. The graph underlying is the swap walk of X, known to have excellent spectral expansion. The swap cosystolic expansion of X is defined to be the cosystolic expansion of . Our main result is a exp(−O(√r)) lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of exp(−O(r)).

Let X be a family of k-element subsets of [n] and let {fs:s→Σ : s∈ X} be an ensemble of local functions, each defined over a subset s⊂ [n]. Is there a global function G:[n]→Σ such that fs = G|s for all s∈ X ? An agreement test is a randomized property tester for this question.

We construct a new explicit family of good quantum low-density parity-check codes which additionally have linear time decoders. Our codes are based on a three-term chain (2m× m)V →δ0 (2m)E →δ1 2F where V (X-checks) are the vertices, E (qubits) are the edges, and F (Z-checks) are the squares of a left-right Cayley complex, and where the maps are defined based on a pair of constant-size random codes CA,CB:2m→2Δ where Δ is the regularity of the underlying Cayley graphs.

A seminal result in learning theory characterizes the PAC learnability of binary classes through the Vapnik-Chervonenkis dimension. Extending this characterization to the general multiclass setting has been open since the pioneering works on multiclass PAC learning in the late 1980s. This work resolves this problem: we characterize multiclass PAC learnability through the DS dimension, a combinatorial dimension defined by Daniely and Shalev-Shwartz, (2014). The classical characterization of the binary case boils down to empirical risk minimization. In contrast, our characterization of the multiclass case involves a variety of algorithmic ideas; these include a natural setting we call list PAC learning. In the list learning setting, instead of predicting a single outcome for a given unseen input, the goal is to provide a short menu of predictions. Our second main result concerns the Natarajan dimension, which has been a central candidate for characterizing multiclass learnability. This dimension was introduced by Natarajan (1988) as a barrier for PAC learning. He furthered showed that it is the only barrier, provided that the number of labels is bounded. Whether the Natarajan dimension characterizes PAC learnability in general has been posed as an open question in several papers since. This work provides a negative answer: we construct a non-learnable class with Natarajan dimension 1. For the construction, we identify a fundamental connection between concept classes and topology (i.e., colorful simplicial complexes). We crucially rely on a deep and involved construction of hyperbolic pseudo-manifolds by Januszkiewicz and Światkowski. It is interesting that hyperbolicity is directly related to learning problems that are difficult to solve although no obvious barriers exist. This is another demonstration of the fruitful links machine learning has with different areas in mathematics.

A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads q bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter q is called the locality of the tester.

We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author~(FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

We strengthen the notion of double samplers, first introduced by Dinur and Kaufman [``High dimensional expanders imply agreement expanders,” in Proc. 58th IEEE Symp. on Foundations of Comp. Science, IEEE, 2017, pp. 974--985], which are samplers with additional combinatorial properties, and whose existence we prove using high-dimensional expanders. The ABNNR code construction [N. Alon et al., IEEE Trans. Inform. Theory, 38 (1992), pp. 509--516] achieves large distance by starting with a base code $C$ with moderate distance, and then amplifying the distance using a sampler. We show that if the sampler is part of a larger double sampler, then the construction has an efficient list-decoding algorithm. Our algorithm works even if the ABNNR construction is not applied to a base code $C$ but rather to any string. In this case the resulting code is approximate-list-decodable, i.e., the output list contains an approximation to the original input. Our list-decoding algorithm works as follows: It uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list-decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results. Double samplers and high-dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.

We introduce a framework of layered subsets, and give a sufficient condition for when a set system supports an agreement test. Agreement testing is a certain type of property testing that generalizes PCP tests such as the plane vs. plane test. Previous work has shown that high dimensional expansion is useful for agreement tests. We extend these results to more general families of subsets, beyond simplicial complexes. These include - Agreement tests for set systems whose sets are faces of high dimensional expanders. Our new tests apply to all dimensions of complexes both in case of two-sided expansion and in the case of one sided partite expansion. This improves and extends an earlier work of Dinur and Kaufman (FOCS 2017) and applies to matroids, and potentially many additional complexes. - Agreement tests for set systems whose sets are neighborhoods of vertices in a high dimensional expander. This family resembles the expander neighborhood family used in the gap-amplification proof of the PCP theorem. This set system is quite natural yet does not sit in a simplicial complex, and demonstrates some versatility in our proof technique. - Agreement tests on families of subspaces (also known as the Grassmann poset). This extends the classical low degree agreement tests beyond the setting of low degree polynomials. Our analysis relies on a new random walk on simplicial complexes which we call the “complement random walk” and which may be of independent interest. This random walk generalizes the non-lazy random walk on a graph to higher dimensions, and has significantly better expansion than previously-studied random walks on simplicial complexes.

We show that every set in P is strongly testable under a suitable encoding. By "strongly testable" we mean having a (proximity oblivious) tester that makes a constant number of queries and rejects with probability that is proportional to the distance of the tested object from the property. By a "suitable encoding" we mean one that is polynomial-time computable and invertible. This result stands in contrast to the known fact that some sets in P are extremely hard to test, providing another demonstration of the crucial role of representation in the context of property testing. The testing result is proved by showing that any set in P has a strong canonical PCP, where canonical means that (for yes-instances) there exists a single proof that is accepted with probability 1 by the system, whereas all other potential proofs are rejected with probability proportional to their distance from this proof. In fact, we show that UP equals the class of sets having strong canonical PCPs (of logarithmic randomness), whereas the class of sets having strong canonical PCPs with polynomial proof length equals "unambiguous- MA". Actually, for the testing result, we use a PCP-of-Proximity version of the foregoing notion and an analogous positive result (i.e., strong canonical PCPPs of logarithmic randomness for any set in UP).

A local tester for an error-correcting code is a probabilistic procedure that queries a small subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is robust if for non-codewords it satisfies the stronger property that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests. In this work we show that for certain codes, any (natural) local tester can be converted to a roubst tester with roughly the same number of queries. Our result holds for the class of affine-invariant lifted codes which is a broad class of codes that includes Reed-Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robustness of lifted codes. To obtain the above transformation we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on expansion / sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in F_q^m, and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views.

We propose a new paradigm for studying the structure of Boolean functions on the biased Boolean hypercube, i.e. when the measure is µp and p is potentially very small, e.g. as small as O(1/n). Our paradigm is based on the following simple fact: the p-biased hypercube is expressible as a convex combination of many small-dimensional copies of the uniform hypercube. To uncover structure for µp, we invoke known structure theorems for µ1/2, obtaining a structured approximation for each copy separately. We then sew these approximations together using a novel “agreement theorem”. This strategy allows us to lift structure theorems from µ1/2 to µp. We provide two applications of this paradigm: Our main application is a structure theorem for functions that are nearly low degree in the Fourier sense. The structure we uncover in the biased hypercube is not at all the same as for the uniform hypercube, despite using the structure theorem for the uniform hypercube as a black box. Rather, new phenomena emerge: whereas nearly low degree functions on the uniform hypercube are close to juntas, when p becomes small, non-juntas arise as well. For example, the function max(y1, · · ·, yε/p) (where yi ∊ {0, 1}) is nearly degree 1 despite not being close to any junta. A second (technically simpler) application is a test for being low degree in the GF(2) sense, in the setting of the biased hypercube. In both cases, we use as a black box the corresponding result for p = 1/2. In the first case, it is the junta theorem of Kindler and Safra, and in the second case, the low degree testing theorem of Alon et al. [IEEE Trans. Inform. Theory, 2005] and Bhattacharyya et al. [Proc. 51st FOCS, 2010]. A key component of our proof is a new local-to-global agreement theorem for higher dimensions, which extends the work of Dinur and Steurer [Proc. 29th CCC, 2014]. Whereas their result sews together vectors, our agreement theorem sews together labeled graphs and hypergraphs. The proof of our agreement theorem uses a novel pruning lemma for hypergraphs, which may be of independent interest. The pruning lemma trims a given hypergraph so that the number of hyperedges in a random induced subhypergraph has roughly a Poisson distribution, while maintaining the expected number of hyperedges.

We develop the notion of double samplers, first introduced by Dinur and Kaufman [DK17], which are samplers with additional combinatorial properties, and whose existence we prove using high dimensional expanders. We show how double samplers give a generic way of amplifying distance in a way that enables efficient list-decoding. There are many error correcting code constructions that achieve large distance by starting with a base code C with moderate distance, and then amplifying the distance using a sampler, e.g., the ABNNR code construction [ABN+ 92] is such. We show that if the sampler is part of a larger double sampler then the construction has an efficient list-decoding algorithm and the list decoding algorithm is oblivious to the base code C (i.e., it runs the unique decoder for C in a black box way). Our list-decoding algorithm works as follows: it uses a local voting scheme from which it constructs a unique games constraint graph. The constraint graph is an expander, so we can solve unique games efficiently. These solutions are the output of the list decoder. This is a novel use of a unique games algorithm as a subroutine in a decoding procedure, as opposed to the more common situation in which unique games are used for demonstrating hardness results. Double samplers and high dimensional expanders are akin to pseudorandom objects in their utility, but they greatly exceed random objects in their combinatorial properties. We believe that these objects hold significant potential for coding theoretic constructions and view this work as demonstrating the power of double samplers in this context.

This special section comprises eleven fully refereed papers whose extended abstracts were presented at the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016) in New Brunswick, New Jersey, October 9--11, 2016. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2016 proceedings. The regular conference program consisted of 85 papers chosen from among 307 submissions. These were selected by a program committee consisting of Eric Blais, Mark Braverman, Siu On Chan, Moses Charikar, Marek Cygan, Irit Dinur (chair), Andrew Drucker, Faith Ellen, Sariel Har-Peled, Prahladh Harsha, Alexandra Kolla, Swastik Kopparty, Robert Krauthgamer, Brendan Lucier, Or Meir, Raghu Meka, Daniele Micciancio, Moni Naor, Joe Neeman, Rasmus Pagh, Rocco Servedio, Yaoyun Shi, Ola Svensson, Thomas Vidick, and Daniel Wichs. The papers invited to this special section were also selected with the input of the program committee. The eleven papers in this section include a broad range of topics, including algorithms, combinatorics, circuit complexity, data structures, learning theory, parameterized complexity, and probability. We thank the authors and the anonymous referees for their efforts. In addition, we would like to thank SICOMP Editor-in-Chief Leonard Schulman and SIAM Senior Publications Coordinator Heather Blythe for their help in preparing this special section.

We initiate the study of Boolean function analysis on high-dimensional expanders. We describe an analog of the Fourier expansion and of the Fourier levels on simplicial complexes, and generalize the FKN theorem to high-dimensional expanders. Our results demonstrate that a high-dimensional expanding complex X can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing |X(k)|=O(n) points in comparison to binom{n}{k+1} points in the (k+1)-slice (which consists of all n-bit strings with exactly k+1 ones).

We study 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph G = (V,E), an alphabet set Sigma and, for each edge {u, v}, a constraint C_uv, the goal is to find an assignment sigma from V to Sigma that satisfies as many constraints as possible, where a constraint C_uv is said to be satisfied by sigma if C_uv contains (sigma(u),sigma(v)). While the approximability of 2-CSPs is quite well understood when the alphabet size |Sigma| is constant (see e.g. [37]), many problems are still open when |Sigma| becomes super constant. One open problem that has received significant attention in the literature is whether it is hard to approximate 2-CSPs to within a polynomial factor of both |Sigma| and |V| (i.e. (|Sigma||V|)^Omega(1) factor). As a special case of the so-called Sliding Scale Conjecture, Bellare et al. [5] suggested that the answer to this question might be positive. Alas, despite many efforts by researchers to resolve this conjecture (e.g. [39, 4, 20, 21, 35]), it still remains open to this day. In this work, we separate |V| and |Sigma| and ask a closely related but weaker question: is it hard to approximate 2-CSPs to within a polynomial factor of |V| (while |Sigma| may be super-polynomial in |Sigma|)? Assuming the exponential time hypothesis (ETH), we answer this question positively: unless ETH fails, no polynomial time algorithm can approximate 2-CSPs to within a factor of |V|^{1-o(1)}. Note that our ratio is not only polynomial but also almost linear. This is almost optimal since a trivial algorithm yields an O(|V|)-approximation for 2-CSPs. Thanks to a known reduction [25, 16] from 2-CSPs to the Directed Steiner Network (DSN) problem, our result implies an inapproximability result for the latter with polynomial ratio in terms of the number of demand pairs. Specifically, assuming ETH, no polynomial time algorithm can approximate DSN to within a factor of k^{1/4 - o(1)} where k is the number of demand pairs. The ratio is roughly the square root of the approximation ratios achieved by best known polynomial time algorithms [15, 26], which yield O(k^{1/2 + epsilon})-approximation for every constant epsilon > 0. Additionally, under Gap-ETH, our reduction for 2-CSPs not only rules out polynomial time algorithms, but also fixed parameter tractable (FPT) algorithms parameterized by the number of variables |V|. These are algorithms with running time g(|V|)·|Sigma|^O(1) for some function g. Similar improvements apply for DSN parameterized by the number of demand pairs k.

We present a polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints. In the “yes” case the fraction of satisfied constraints is at least 1 −ε, and in the “no” case we show that this fraction is at most ε, assuming a certain (new) combinatorial hypothesis on the Grassmann graph. In other words, we describe a combinatorial hypothesis that implies the 2-to-1 conjecture with imperfect completeness. The companion submitted paper [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] makes some progress towards proving this hypothesis.

Given a function f:[N]^k->[M]^k, the Z-test is a three query test for checking if a function f is a direct product, namely if there are functions g_1,...g_k:[N]->[M] such that f(x_1,...,x_k)=(g_1(x_1),...,g_k(x_k)) for every input x in [N]^k. This test was introduced by Impagliazzo et. al. (SICOMP 2012), who showed that if the test passes with probability epsilon > exp(-sqrt k) then f is Omega(epsilon) close to a direct product function in some precise sense. It remained an open question whether the soundness of this test can be pushed all the way down to exp(-k) (which would be optimal). This is our main result: we show that whenever f passes the Z test with probability epsilon > exp(-k), there must be a global reason for this: namely, f must be close to a product function on some Omega(epsilon) fraction of its domain. Towards proving our result we analyze the related (two-query) V-test, and prove a "restricted global structure" theorem for it. Such theorems were also proven in previous works on direct product testing in the small soundness regime. The most recent work, by Dinur and Steurer (CCC 2014), analyzed the V test in the exponentially small soundness regime. We strengthen their conclusion of that theorem by moving from an "in expectation" statement to a stronger "concentration of measure" type of statement, which we prove using hyper-contractivity. This stronger statement allows us to proceed to analyze the Z test. We analyze two variants of direct product tests. One for functions on ordered tuples, as above, and another for functions on sets of size k. The work of Impagliazzo et al. was actually focused only on functions of the latter type, i.e. on sets. We prove exponentially small soundness for the Z-test for both variants. Although the two appear very similar, the analysis for tuples is more tricky and requires some additional ideas.

We show that high dimensional expanders imply derandomized direct product tests, with a number of subsets that is linear in the size of the universe. Direct product tests belong to a family of tests called agreement tests that are important components in PCP constructions and include, for example, low degree tests such as line vs. line and plane vs. plane. For a generic hypergraph, we introduce the notion of agreement expansion, which captures the usefulness of the hypergraph for an agreement test. We show that explicit bounded degree agreement expanders exist, based on Ramanujan complexes.

We revisit the Raz-Safra plane-vs.-plane test and study the closely related cube vs. cube test. In this test the tester has access to a "cubes table" which assigns to every cube a low degree polynomial. The tester randomly selects two cubes (affine sub-spaces of dimension 3) that intersect on a point x in F^m, and checks that the assignments to the cubes agree with each other on the point x. Our main result is a new combinatorial proof for a low degree test that comes closer to the soundness limit, as it works for all epsilon >= poly(d)/{|F|}^{1/2}, where d is the degree. This should be compared to the previously best soundness value of epsilon >= poly(m, d)/|F|^{1/8}. Our soundness limit improves upon the dependence on the field size and does not depend on the dimension of the ambient space. Our proof is combinatorial and direct: unlike the Raz-Safra proof, it proceeds in one shot and does not require induction on the dimension of the ambient space. The ideas in our proof come from works on direct product testing which are even simpler in the current setting thanks to the low degree. Along the way we also prove a somewhat surprising fact about connection between different agreement tests: it does not matter if the tester chooses the cubes to intersect on points or on lines: for every given table, its success probability in either test is nearly the same.

We investigate the value of parallel repetition of one-round games with any number of players k>=2. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games. Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are *connected* in a certain sense. We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question.

The $k$-fold direct sum encoding of a string $a \in \{0,1\}^n$ is a function $f_a$ that takes as input sets $S \subseteq [n]$ of size $k$ and outputs $f_a(S) = \sum_{i \in S} a_i \pmod 2$. In this paper we prove a direct sum testing theorem. We describe a three query test that accepts with probability one any function of the form $f_a$ for some $a$ and rejects with probability $\Omega(\varepsilon)$ functions $f$ that are $\varepsilon$-far from being a direct sum encoding, where the constant behind the $\Omega$ notation is independent of $k$. This theorem has a couple of additional guises: Linearity testing: By identifying the subsets of $[n]$ with vectors in $\{0,1\}^n$ in the natural way, our result can be thought of as a linearity testing theorem for functions whose domain is restricted to the $k$th layer of the hypercube (i.e., the set of $n$-bit strings with Hamming weight $k$). Tensor power testing: By moving to $-1,1$ notation, the direct sum encoding is equivalent (up to a difference thatis negligible when $k\ll \sqrt n$) to a tensor power. Thus our theorem implies a three query test for deciding if a given tensor $f\in \{-1,1\}^{n^k}$ is a tensor power of a single dimensional vector $a\in \{-1,1\}^n$, i.e., whether there is some $a$ such that $f = a^{\otimes k}$. We also provide a four query test for checking if a given $\pm 1$ matrix has rank $1$. Our test naturally extends the linearity test of Blum, Luby, and Rubinfeld [ J. Comput. Syst. Sci., 47 (1993), pp. 549--595]. Our analysis proceeds by first handling the $k=n/2$ case and then reducing this case to the general $k

One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de-Morgan formulas. Karchmer, Raz, and Wigderson suggested to approach this problem by proving that formula complexity behaves "as expected" with respect to the composition of functions f * g. They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds. The first step toward proving the KRW conjecture was made by Edmonds et al., who proved an analogue of the conjecture for the composition of "universal relations". In this work, we extend the argument of Edmonds et al. further to f * g where f is an arbitrary function and g is the parity function. While this special case of the KRW conjecture was already proved implicitly in Hastad's work on random restrictions, our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer and Wigderson. In addition, our proof gives a new structural result, which roughly says that the naive way for computing f * g is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n^{3-o(1)} due to Hastad.

The k-fold direct sum encoding of a string α ∈ --0,1}n is a function fα that takes as input sets S ⊆ [n] of size k and outputs fα (S) = ∑i ∈ S αi (mod 2. In this paper we prove a Direct Sum Testing theorem. We describe a three query test that accepts with probability one any function of the form fα for some α, and rejects with probability Ω(ε) functions f that are ε being a direct sum encoding.

The starting point of this paper is that instances of computational problems often do not exist in isolation. Rather, multiple and correlated instances of the same problem arise naturally in the real world. The challenge is how to gain computationally from instance correlations when they exist. We will be interested in settings where significant computational gain can be made in solving a single primary instance by having access to additional auxiliary instances which are correlated to the primary instance via the solution space.

In this paper, we address the question of whether the recent derandomization results obtained by the use of the low-degree long code can be extended to other product settings. We consider two settings: (1) the graph product results of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of approximate graph coloring. In our first result, we show that there exists a considerably smaller subgraph of $K_3^{\otimes R}$ which exhibits the following property (shown for $K_3^{\otimes R}$ by Alon et al.): independent sets close in size to the maximum independent set are well approximated by dictators. The "majority is stablest" type of result of Dinur et al. and Dinur and Shinkar shows that if there exist two sets of vertices $A$ and $B$ in $K_3^{\otimes R}$ with very few edges with one endpoint in $A$ and another in $B$, then it must be the case that the two sets $A$ and $B$ share a single influential coordinate. In our second result, we show that a similar "majority is stablest" statement holds good for a considerably smaller subgraph of $K_3^{\otimes R}$. Furthermore using this result, we give a more efficient reduction from Unique Games to the graph coloring problem, leading to improved hardness of approximation results for coloring.

We show that every language in NP has a PCP verifier that tosses O(log n) random coins, has perfect completeness, and a soundness error of at most 1/poly(n), while making O(poly log log n) queries into a proof over an alphabet of size at most n1/poly log log n. Previous constructions that obtain 1/poly(n) soundness error used either poly log n queries or an exponential alphabet, i.e. of size 2nc for some c> 0. Our result is an exponential improvement in both parameters simultaneously. Our result can be phrased as polynomial-gap hardness for approximate CSPs with arity poly log log n and alphabet size n1/poly log n. The ultimate goal, in this direction, would be to prove polynomial hardness for CSPs with constant arity and polynomial alphabet size (aka the sliding scale conjecture for inverse polynomial soundness error).

A direct product function is a function of the form g(x1, ⋯, xk)=(g1(x1), ⋯, g(xk)). We show that the direct product property is locally testable with two queries, that is, a canonical two-query test distinguishes between direct product functions and functions that are far from direct products with constant probability. This local testing question comes up naturally in the context of PCPs, where direct products play a prominent role for gap amplification. We consider the following natural two query test for a given function f:[N]k→[M]kTwo query direct product test: Choose x, y that agree on a random set A of t coordinates and accept if f(x)A=f(y)A. We provide a comprehensive analysis of this test for all parameters N, M, k, t≤O(k) and success probability δ>0. Our main result is that if a given function f:[N]k→[M]kpasses the test with probability δ≥1-ε then there is a direct product function g such that P[f(x)=g(x)]≥1-O(ε). This is the first result relating success in the above (or any) test to the fraction of the domain on which f is equal to a direct product function. This test has been analyzed in previous works for the case t≪k≪N, and results show closeness of f to a direct product under a less natural measure of "approximate agreement". In the small soundness regime, we prove that if the test above passes with probability δ ≥ exp(-k), then the function agrees with a direct product function on local parts of the domain. This extends the previous range of parameters of δ≥exp(-3√k) to the entire meaningful range of δ>exp(-k).

We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game G of entangled value 1 - εc)k), for some universal constant c ≥ 1. Previously parallel repetition with an exponential decay in k was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework recently introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games. Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call “vector quantum strategy” to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, rounding relies on a quantum analogue of Holenstein's correlated sampling lemma which may be of independent interest. Our “quantum correlated sampling lemma” generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two isolated parties, given classical descriptions of arbitrary bipartite states |ψ〉, |φ〉 respectively such that |ψ〉 ≈ |φ〉, are able to locally generate a joint entangled state|Ψ〉 ≈ |ψ〉 ≈ |φ〉 using an initial entangled state that is independent of their inputs.

We propose an analytical framework for studying parallel repetition, a basic product operation for one-round twoplayer games. In this framework, we consider a relaxation of the value of projection games. We show that this relaxation is multiplicative with respect to parallel repetition and that it provides a good approximation to the game value. Based on this relaxation, we prove the following improved parallel repetition bound: For every projection game G with value at most ρ, the k-fold parallel repetition G⊗k has value at most

We study the covering complexity of constraint satisfaction problems (CSPs). The covering number of a CSP instance C, denoted v(C), is the smallest number of assignments to the variables, such that each constraint is satisfied by at least one of the assignments. This covering notion describes situations in which we must satisfy all the constraints, and are willing to use more than one assignment to do so. At the same time, we want to minimize the number of assignments. We study the covering problem for different constraint predicates. We first observe that if the predicate contains an odd predicate, then it is covered by any assignment and its negation. In particular, 3CNF and 3LIN, that are hard in the max-CSP sense, are easy to cover. However, the covering problem is hard for predicates that do not contain an odd predicate: 1. For the 4LIN predicate, it is NP-hard to decide if a given instance C has v(C) at most 2, or v(C) is super-constant. 2. (a) We propose a framework of covering dictatorship tests. We design and analyze such a dictatorship test for every predicate that supports a pair wise independent distribution. (b) We introduce a covering unique games conjecture, and use it to convert the covering dictatorship tests into conditional hardness results. 3. Finally, we study a hypothesis about the hardness of covering random instances that is similar to Feige's R3SAT hypothesis. We show the following somewhat surprising implication: If our hypothesis holds for dense enough instances, then it is hard to color an O(1)-colorable hyper graph with a polynomial number of colors.

We develop new techniques to incorporate the recently proposed “short code” (a low-degree version of the long code) into the construction and analysis of PCPs in the classical “Label Cover + Fourier Analysis” framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs and certain coloringtype problems. In particular, we show a hardness for a variant of hypergraph coloring (with hyperedges of size 6), with a gap between 2 and exp(2Ω(√log log N)) number of colors where N is the number of vertices. This is the first hardness result to go beyond the O(log N) barrier for a coloring-type problem. Our hardness bound is a doubly exponential improvement over the previously known O(log log N)-coloring hardness for 2-colorable hypergraphs, and an exponential improvement over the (logN)Ω(1)-coloring hardness for O(1)-colorable hypergraphs. Stated in terms of “covering complexity,” we show that for 6-ary Boolean CSPs, it is hard to decide if a given instance is perfectly satisfiable or if it requires more than 2Ω(√log log N) assignments for covering all of the constraints. While our methods do not yield a result for conventional hypergraph coloring due to some technical reasons, we also prove hardness of (log N)Ω(1)-coloring 2-colorable 6-uniform hypergraphs (this result relies just on the long code). A key algebraic result driving our analysis concerns a very low-soundness error testing method for Reed-Muller codes. We prove that if a function β : F2m→ F2is 2Ω(d)far in absolute distance from polynomials of degree m-d, then the probability that deg(βg) ≤ m-3d/4 for a random degree d/4 polynomial g is doubly exponentially small in d.

The main result of this paper is a generic composition theorem for low-error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well understood only in the constant error regime. Existing composition methods in the low-error regime were nonmodular (i.e., very much tailored to the specific PCPs that were being composed), resulting in complicated constructions of PCPs. Furthermore, until recently, composition in the low-error regime suffered from incurring an extra “consistency” query, resulting in PCPs that are not “two-query” and hence, much less useful for hardness-of-approximation reductions. In a recent breakthrough, Moshkovitz and Raz (Proceedings of the 49th IEEE Symposium on Foundations of Computer Science (FOCS), 2008) [J. ACM, 57 (2010)] constructed almost linear-sized low-error 2-query PCPs for every language in NP. Indeed, the main technical component of their construction is a novel composition of certain specific PCPs. We generalize and abstract their composition method, thereby giving a modular and simpler proof of their result. To facilitate the modular composition, we introduce a new variant of PCP, which we call a decodable PCP (dPCP). A dPCP is an encoding of an NP witness that is both locally checkable and locally decodable. The dPCP verifier, in addition to verifying the validity of the given proof like a standard PCP verifier, also locally decodes the original NP witness. Our composition is generic in the sense that it works regardless of the way the component PCPs are constructed.

This paper strengthens the low-error PCP characterization of NP, coming closer to the upper limit of the BGLR conjecture. Consider the task of verifying a written proof for the membership of a given input in an NP language. In this paper, this is achieved by making a constant number of accesses to the proof, obtaining error probability that is exponentially small in the total number of bits that are read. We show that the number of bits that are read in each access to the proof can be made as high as log β n , for any constant β < 1, where n is the length of the proof. The BGLR conjecture asserts the same for any constant β, for β smaller or equal to 1. Our results are in fact stronger, implying that the Gap-Quadratic-Solvability problem with a constant number of variables in each equation is NP-hard. That is, given a system of n quadratic equations over a field $${\mathcal{F}}$$ of size up to $$2^{\log^\beta n}$$ , where each equation depends on a constant number of variables, it is NP-hard to distinguish between the case where there is a common solution to all of the equations and the case where any assignment satisfies at most a $${2 / |\mathcal{F}|}$$ fraction of them. At the same time, our proof presents a direct construction of a low-degree test whose error-probability is exponentially small in the number of bits accessed. Such a result was previously known only relying on recursive applications of the entire PCP theorem.

A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof and in return is allowed to err with some bounded probability. The probability that the verifier accepts a proof of a false claim is called the soundness error and is an important parameter of a PCP system that one seeks to minimize. Constructing PCPs with subconstant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability. In this work, we construct such PCP verifiers, i.e., PCPs that make only two queries and have subconstant soundness error. Our construction can be viewed as a combinatorial alternative to the “manifold vs. point” construction, which is the basis for all constructions in the literature for this parameter range. The “manifold vs. point” PCP is based on a low-degree test, while our construction is based on a direct product test. We also extend our construction to yield a decodable PCP (dPCP) with the same parameters. By plugging in this dPCP into the scheme of Dinur and Harsha (FOCS 2009), one gets an alternative construction of the result of Moshkovitz and Raz (FOCS 2008), namely a construction of two-query PCPs with small soundness error and small alphabet size. Our construction of a PCP is based on extending the derandomized direct product test of Impagliazzo, Kabanets, and Wigderson (STOC 09) to a derandomized parallel repetition theorem. More accurately, our PCP construction is obtained in two steps. We first prove a derandomized parallel repetition theorem for specially structured PCPs. Then, we show that any PCP can be transformed into one that has the required structure, by embedding it on a de-Bruijn graph.

A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof, and in return is allowed to err with some bounded probability. The probability that the verifier accepts a false proof is called the soundness error, and is an important parameter of a PCP system that one seeks to minimize. Constructing PCPs with sub-constant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability. In this work we construct such PCP verifiers, i.e., PCPs that make only two queries and have sub-constant soundness error. Our construction can be viewed as a combinatorial alternative to the "manifold vs. point'' construction, which is the only construction in the literature for this parameter range. The "manifold vs. point'' PCP is based on a low degree test, while our construction is based on a direct product test. Our construction of a PCP is based on extending the derandomized direct product test of Impagliazzo, Kabanets and Wigderson (STOC 09) to a derandomized parallel repetition theorem. More accurately, our PCP construction is obtained in two steps. We first prove a derandomized parallel repetition theorem for specially structured PCPs. Then, we show that any PCP can be transformed into one that has the required structure, by embedding it on a de-Bruijn graph.

For every ∈ > 0, and integer q ≥ 3, we show that given an N-vertex graph that has an induced q-colorable subgraph of size (1 - ∈)N, it is NP-hard to find an independent set of size N/q2.

The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs). Prior to this work, composition of PCPs was well-understood only in the constant error regime. Existing composition methods in the low error regime were non-modular (i.e., very much tailored to the specific PCPs that were being composed), resulting in complicated constructions of PCPs. Furthermore, until recently, composition in the low error regime suffered from incurring an extra 'consistency' query, resulting in PCPs that are not 'two-query' and hence, much less useful for hardness-of-approximation reductions. In a recent breakthrough, Moshkovitz and Raz [In Proc. 49th IEEE Symp. on Foundations of Comp. Science (FOCS), 2008] constructed almost linear-sized low-error 2-query PCPs for every language in NP. Indeed, the main technical component of their construction is a novel composition of certain specific PCPs. We give a modular and simpler proof of their result by repeatedly applying the new composition theorem to known PCP components. To facilitate the new modular composition, we introduce a new variant of PCP, which we call a "decodable PCP (dPCP)". A dPCP is an encoding of an NP witness that is both locally checkable and locally decodable. The dPCP verifier in addition to verifying the validity of the given proof like a standard PCP verifier, also locally decodes the original NP witness. Our composition is generic in the sense that it works regardless of the way the component PCPs are constructed.

We study the AprxColoring$(q,Q)$ problem: Given a graph G, decide whether $\chi(G)\le q$ or $\chi(G)\ge Q$. We present hardness results for this problem for any constants $3\le q

Given a function f : X rarr Sigma, its lscr-wise direct product is the function F = flscr: Xlscrrarr Sigmalscrdefined by: F(x1,...,xlscr) = (f(x1),...,f(xlscr)). We are interested in the local testability of the direct product encoding (mapping f rarr flscr). Namely, given an arbitrary function F : Xlscrrarr Sigmalscr, we wish to determine how close it is to flscrfor some f : X rarr Sigma, by making two random queries into F. In this work we analyze the case of low acceptance probability of the test. We show that even if the test passes with small probability, epsiv>0, already F must have a non-trivial structure and in particular must agree with some flscron nearly epsiv of the domain. Moreover, we give a structural characterization of all functions F on which the test passes with probability epsiv. Our results can be viewed as a combinatorial analog of the low error dasialow degree testpsila, that is used in PCP constructions.

Given a pair of finite groups G and H, the set of homomorphisms from G to H form an error-correcting code where codewords differ in at least 1/2 the coordinates. We show that for every pair of abelian groups G and H, the resulting code is (locally) list-decodable from a fraction of errors arbitrarily close to its distance. At the heart of this result is the following combinatorial result: There is a fixed polynomial p(•) such that for every pair of abelian groups G and H, if the maximum fraction of agreement between two distinct homomorphisms from G to H is Λ, then for every ε> 0 and every function f:G -> H, the number of homomorphisms that have agreement Λ + ε with f is at most p(1/ε). We thus give a broad class of codes whose list-decoding radius exceeds the "Johnson bound". Examples of such codes are rare in the literature, and for the ones that do exist, "combinatorial" techniques to analyze their list-decodability are limited. Our work is an attempt to add to the body of such techniques. We use the fact that abelian groups decompose into simpler ones and thus codes derived from homomorphisms over abelian groups may be viewed as certain "compositions" of simpler codes. We give techniques to lift list-decoding bounds for the component codes to bounds for the composed code. We believe these techniques may be of general interest.

The PCP theorem [Arora and Safra 1998; Arora et. al. 1998] says that every language in NP has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof. The celebrated equivalence of this theorem and inapproximability of certain optimization problems, due to Feige et al. [1996], has placed the PCP theorem at the heart of the area of inapproximability. In this work, we present a new proof of the PCP theorem that draws on this equivalence. We give a combinatorial proof for the NP-hardness of approximating a certain constraint satisfaction problem, which can then be reinterpreted to yield the PCP theorem. Our approach is to consider the unsat value of a constraint system, which is the smallest fraction of unsatisfied constraints, ranging over all possible assignments for the underlying variables. We describe a new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system. The amplification step causes an increase in alphabet-size that is corrected by a (standard) PCP composition step. Iterative application of these two steps yields a proof for the PCP theorem. The amplification lemma relies on a new notion of “graph powering” that can be applied to systems of binary constraints. This powering amplifies the unsat-value of a constraint system provided that the underlying graph structure is an expander. We also extend our amplification lemma towards construction of assignment testers (alternatively, PCPs of Proximity) which are slightly stronger objects than PCPs. We then construct PCPs and locally-testable codes whose length is linear up to a polylog factor, and whose correctness can be probabilistically verified by making a constant number of queries. Namely, we prove SAT ∈ PCP 1/2,1 [log 2 ( n ⋅poly log n ), O (1)].

A theorem of Bourgain [4] on Fourier tails states that if f :(-1, 1)n → (-1, 1) is a boolean-valued function on the discrete cube such that for any k > 0, [Σ|S| > k f(S)2 < k-1/2 + o(1), ] then essentially, f depends on only 2O(k) coordinates. This and related theorems such as Friedgut's Theorem [12], KKL [16], the FKN Theorem [14], and the Majority Is Stablest Theorem [27] have proven useful for numerous results in theoretical computer science [3, 5, 9, 6, 7, 10, 11, 18, 19, 20, 24, 17, 25, 23, 22, 28, 29, 31].In this paper we prove an analogue to Bourgain's Theorem for bounded functions on the discrete cube, f : (n ⋺ [-1,1]); such functions arise naturally in hardness-of-approximation problems, as averages of boolean functions. Specifically, we show that for every k > 0, if [Σ|S| > k f(S)2 < exp(-O(k2 log k))] then essentially, f depends on only 2O(k) coordinates. We also show, perhaps surprisingly, that this result is sharp up to the log k factor in the exponent.Our proof uses Fourier analysis, as well as some extremal properties of the Chebyshev polynomials.

We study the APPROXCOLORING q(Q) problem: Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q. We derive conditional hardness for this problem for any constant 3 ≤ q < Q. For q ≥ 4, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For q=3, we base our hardness result on a certain 'fish shaped' variant of his conjecture.We also prove that the problem ALMOST-3-COLORINGε is hard for any constant ε>0, assuming Khot's Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a ε fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least ε.Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO'05].

In this work we look back into the proof of the PCP (probabilistically checkable proofs) theorem, with the goal of finding new proofs that are “more combinatorial” and arguably simpler. For that we introduce the notion of an assignment tester, which is a strengthening of the standard PCP verifier, in the following sense. Given a statement and an alleged proof for it, while the PCP verifier checks correctness of the statement, the assignment tester checks correctness of the statement and the proof. This notion enables composition that is truly modular; i.e., one can compose two assignment testers without any assumptions on how they are constructed. A related notion called PCPs of proximity was independently introduced in [E. Ben‐Sasson et al., Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, 2004, ACM, New York, 2004, pp. 1–10]. We provide a toolkit of (nontrivial) generic transformations on assignment testers. These transformations may be interesting in their own right, and allow us to present the following two main results: 1. A new proof of the PCP theorem. This proof relies on a rather weak assignment tester given as a “black box.” From this, we construct combinatorially the full PCP. An important component of this proof is a new combinatorial aggregation technique (i.e., a new transformation that allows the verifier to read fewer, though possibly longer, “pieces” of the proof). An implementation of the black‐box tester can be obtained from the algebraic proof techniques that already appear in [L. Babai et al., Proceedings of the 23rd ACM Symposium on Theory of Computing, New Orleans, LA, 1991, ACM, New York, 1991, pp. 21–31; U. Feige et al., J. ACM, 43 (1996), pp. 268–292]. 2. Our second construction is a “standalone” combinatorial construction showing $NP \subseteq PCP[polylog, 1]$. This implies, for example, that approximating max‐SAT is quasi‐NP‐hard. This construction relies on a transformation that makes an assignment tester “oblivious,” so that the proof locations read are independent of the statement that is being proven. This eliminates, in a rather surprising manner, the need for aggregation in a crucial point in the proof.

Given a k-uniform hypergraph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyperedge. We present a new multilayered probabilistically checkable proof (PCP) construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within a factor of $(k-1-\epsilon)$ for arbitrary constants $\epsilon>0$ and $k\ge 3$. The result is nearly tight as this problem can be easily approximated within factor k. Our construction makes use of the biased long-code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets. We also give a different proof that shows an inapproximability factor of $\lfloor \frac{k}{2} \rfloor -\eps$. In addition to being simpler, this proof also works for superconstant values of k up to (log N) 1/c , where c > 1 is a fixed constant and N is the number of hyperedges.

In this work, we look back into the proof of the PCP theorem, with the goal of finding new proofs that are "more combinatorial" and arguably simpler. For that, we introduce the notion of an assignment tester, which is a strengthening of the standard PCP verifier, in the following sense. Given a statement and an alleged proof for it, while the PCP verifier checks correctness of the statement, the assignment-tester checks correctness of the statement and the proof. This notion enables composition that is truly modular, i.e., one can compose two assignment-testers without any assumptions on how they are constructed. A related notion was independently introduced in (Ben-Sasson et. al. STOC 04). We provide a toolkit of (non-trivial) generic transformations on assignment testers. These transformations may be interesting in their own right, and allow us to present the following two main results: 1. The first is a new proof of the PCP theorem. This proof relies on a rather weak assignment tester given as a "black box". From this, we construct combinatorially the full PCP. An important component of this proof is a new combinatorial aggregation technique (i.e., a new transformation that allows the verifier to read fewer, though possibly longer, "pieces" of the proof). An implementation of the black-box tester can be obtained from the algebraic proof techniques that already appear in L. Babai et al., 1991 and U. Feige et al., 1991. Obtaining a combinatorial implementation of this tester would give a purely combinatorial proof for the PCP theorem, which we view as an interesting open problem. 2. Our second construction is a "standalone" combinatorial construction showing NP /spl sube/ PCP (S. Arora et al., 1998). This implies, for example, that approximating max-SAT is quasi-NP-hard. This construction relies on a transformation that makes an assignment tester "oblivious": so that the proof locations read are independent of the statement that is being proven. This eliminates, in a rather surprising manner, the need for aggregation in a crucial point in the proof.

We examine the tradeoff between privacy and usability of statistical databases. We model a statistical database by an n-bit string d1,..,dn, with a query being a subset q ⊆ [n] to be answered by Σiεq di. Our main result is a polynomial reconstruction algorithm of data from noisy (perturbed) subset sums. Applying this reconstruction algorithm to statistical databases we show that in order to achieve privacy one has to add perturbation of magnitude (Ω√n). That is, smaller perturbation always results in a strong violation of privacy. We show that this result is tight by exemplifying access algorithms for statistical databases that preserve privacy while adding perturbation of magnitude Õ(√n).For time-T bounded adversaries we demonstrate a privacypreserving access algorithm whose perturbation magnitude is ≈ √T.

Given a k-uniform hyper-graph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within factor (k-1-ε) for any k ≥ 3 and any ε>0. The result is essentially tight as this problem can be easily approximated within factor k. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets.

We prove that coloring a 3-uniform 2-colorable hypergraph with any constant number of colors is NP-hard. The best known algorithm (Krivelevich, Nathaniel, and Sudakov, 2001)colors such a graph using O(n/sup 1/5/) colors. Our result immediately implies that for any constants k > 2 and c/sub 2/ > c/sub 1/ > 1, coloring a k-uniform c/sub 1/-colorable hypergraph with c/sub 2/ colors is NP-hard; leaving completely open only the k = 2 graph case. We are the first to obtain a hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k /spl ges/ 4 such a result has been shown by Guruswami et al. (2000), who also discussed the inherent difference between the k = 3 case and k /spl ges/ 4. Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph (Kneser, 1955; Lovasz, 1978) and the Schrijver graph (Schrijver, 1978). We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has 'many' non-monochromatic edges.

(MATH) We show that the Minimum Vertex Cover problem is NP-hard to approximate to within any factor smaller than $10\sqrt{5}-21 \approx 1.36067$, improving on the previously known hardness result for a $\frac{7}{6}$ factor.

This paper strengthens the law-error PCP characterization of NP, coming closer to the upper limit of the BGLR conjecture.Namely, we prove that witnesses for membership in any NP language can be verified with a constant nunbcr of accesses, and with an error probability exponentially small in the number of bits accessed, where this number is as high as lagan, for any constant fl < 1. (The BGLR conjecture claims the same for any p 5 1).Our results are in fact stronger, implying the Gap-Quadratic-Solvability problem to be NP-hard even if the equations are restricted to having a constant number of variables.That is, given a system of quadratic-equations over a field 3 (of size up to ZLogD"), where each equation depends on a constant number of variables, it is NP-hard to decide between the case where there is a common solution for all of the equations, and the case where any assignment satisfies no more than a & fraction of them.At the same time, ow proof presents a direct eonstmction of a low-degree-test whose error-probability is expancntially small in the number of hits accessed.Such a result was previously known only relying on recursive applications of the entire PCP theorem.

This paper shows the closest vector in a lattice to be NP-hard to approximate to within any factor up to 2/sup (logn)1-4/ where /spl epsiv/=(loglogn)/sup -c/ for any constant c< 1/2.