arXiv subjects

Irit Dinur

Publications and source records attributed to Irit Dinur.

33 records · Page 2Linked to original sources

arXiv2020

Locally testable codes via high-dimensional expanders

Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are "far" from all codewords by probing a given word only at a very few (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. A major open problem is whether there exist LTCs with positive rate, constant relative distance and testable with a constant number of queries. In this paper, we present a new approach towards constructing such LTCs using the machinery of high-dimensional expanders. To this end, we consider the Tanner representation of a code, which is specified by a graph and a base code. Informally, our result states that if this graph is part of a high-dimensional expander then the local testability of the code follows from the local testability of the base code. This work unifies and generalizes the known results on testability of the Hadamard, Reed-Muller and lifted codes on the Subspace Complex, all of which are proved via local self correction. However, unlike previous results, constant rounds of self correction do not suffice as the diameter of the underlying test graph can be logarithmically large in a high-dimensional expander and not constant as in all known earlier results. We overcome this technical hurdle by performing iterative self correction with logarithmically many rounds and tightly controlling the error in each iteration using properties of the high-dimensional expander. Given this result, the missing ingredient towards constructing a constant-query LTC with positive rate and constant relative distance is an instantiation of a base code that interacts well with a constant-degree high-dimensional expander.

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arXiv2019

Direct Sum Testing: The General Case

A function $f:[n_1]\times\dots\times[n_d]\to\mathbb{F}_2$ is a direct sum if it is of the form $f\left(a_1,\dots,a_d\right) = f_1(a_1)\oplus\dots \oplus f_d (a_d),$ for some $d$ functions $f_i:[n_i]\to\mathbb{F}_2$ for all $i=1,\dots, d$, and where $n_1,\dots,n_d\in\mathbb{N}$. We present a $4$-query test which distinguishes between direct sums and functions that are far from them. The test relies on the BLR linearity test (Blum, Luby, Rubinfeld, 1993) and on an agreement test which slightly generalizes the direct product test (Dinur, Steurer, 2014). In multiplicative $\pm 1$ notation, our result reads as follows. A $d$-dimensional tensor with $\pm 1$ entries is called a tensor product if it is a tensor product of $d$ vectors with $\pm 1$ entries, or equivalently, if it is of rank $1$. The presented tests can be read as tests for distinguishing between tensor products and tensors that are far from being tensor products. We also present a different test, which queries the function at most $(d+2)$ times, but is easier to analyze.

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arXiv2019

Near Coverings and Cosystolic Expansion -- an example of topological property testing

We study the stability of covers of simplicial complexes. Given a map $f:Y\to X$ that satisfies almost all of the local conditions of being a cover, is it close to being a genuine cover of $X$? Complexes $X$ for which this holds are called cover-stable. We show that this is equivalent to $X$ being a cosystolic expander with respect to non-abelian coefficients. This gives a new combinatorial-topological interpretation to cosystolic expansion which is a well studied notion of high dimensional expansion. As an example, we show that the $2$-dimensional spherical building $A_{3}(\mathbb{F}_q)$ is cover-stable. We view this work as a possibly first example of "topological property testing", where one is interested in studying stability of a topological notion that is naturally defined by local conditions.

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arXiv2019

Agreement testing theorems on layered set systems

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.

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arXiv2018

ETH-Hardness of Approximating 2-CSPs and Directed Steiner Network

We study the 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph $G=(V,E)$, an alphabet set $Σ$ and, for each $\{u, v\}\in E$, a constraint $C_{uv} \subseteq Σ\timesΣ$, the goal is to find an assignment $σ: V \to Σ$ that satisfies as many constraints as possible, where a constraint $C_{uv}$ is satisfied if $(σ(u),σ(v))\in C_{uv}$. While the approximability of 2-CSPs is quite well understood when $|Σ|$ is constant, many problems are still open when $|Σ|$ becomes super constant. One such problem is whether it is hard to approximate 2-CSPs to within a polynomial factor of $|Σ| |V|$. Bellare et al. (1993) suggested that the answer to this question might be positive. Alas, despite efforts to resolve this conjecture, it remains open to this day. In this work, we separate $|V|$ and $|Σ|$ and ask a related but weaker question: is it hard to approximate 2-CSPs to within a polynomial factor of $|V|$ (while $|Σ|$ may be super-polynomial in $|V|$)? Assuming the exponential time hypothesis (ETH), we answer this question positively by showing that no polynomial time algorithm can approximate 2-CSPs to within a factor of $|V|^{1 - o(1)}$. Note that our ratio is almost linear, which is almost optimal as a trivial algorithm gives a $|V|$-approximation for 2-CSPs. Thanks to a known reduction, our result implies an ETH-hardness of approximating Directed Steiner Network with ratio $k^{1/4 - o(1)}$ where $k$ is the number of demand pairs. The ratio is roughly the square root of the best known ratio achieved by polynomial time algorithms (Chekuri et al., 2011; Feldman et al., 2012). Additionally, under Gap-ETH, our reduction for 2-CSPs not only rules out polynomial time algorithms, but also FPT algorithms parameterized by $|V|$. Similar statement applies for DSN parameterized by $k$.

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arXiv2017

Multiplayer parallel repetition for expander games

We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. 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.

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arXiv2016

Cube vs. Cube Low Degree Test

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 $x\in \mathbf{F}^m$, 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 $ε\ge poly(d)/{\mathbf{F}}^{1/2}$, where $d$ is the degree. This should be compared to the previously best soundness value of $ε\ge poly(m, d)/\mathbf{F}^{1/8}$. 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.

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arXiv2015

Polynomially Low Error PCPs with polyloglog n Queries via Modular Composition

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 $1/\text{poly}(n)$, while making at most $O(\text{poly}\log\log n)$ queries into a proof over an alphabet of size at most $n^{1/\text{poly}\log\log n}$. Previous constructions that obtain $1/\text{poly}(n)$ soundness error used either $\text{poly}\log n $ queries or an exponential sized alphabet, i.e. of size $2^{n^c}$ for some $c>0$. Our result is an exponential improvement in both parameters simultaneously. Our result can be phrased as a polynomial-gap hardness for approximate CSPs with arity $\text{poly}\log\log n$ and alphabet size $n^{1/\text{poly}\log n}$. 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). Our construction is based on a modular generalization of previous PCP constructions in this parameter regime, which involves a composition theorem that uses an extra `consistency' query but maintains the inverse polynomial relation between the soundness error and the alphabet size. Our main technical/conceptual contribution is a new notion of soundness, which we refer to as {\em distributional soundness}, that replaces the previous notion of "list decoding soundness", and that allows us to prove a modular composition theorem with tighter parameters. This new notion of soundness allows us to invoke composition a super-constant number of times without incurring a blow-up in the soundness error.

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arXiv2015

A parallel repetition theorem for entangled projection games

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-eps < 1, the value of the $k$-fold repetition of G goes to zero as O((1-eps^c)^k), for some universal constant c\geq 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.

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arXiv2015

Derandomized Graph Product Results using the Low Degree Long Code

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.

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arXiv2014

Analytical Approach to Parallel Repetition

We propose an analytical framework for studying parallel repetition, a basic product operation for one-round two-player games. In this framework, we consider a relaxation of the value of a game, $\mathrm{val}_+$, and prove that for projection games, it is both multiplicative (under parallel repetition) and a good approximation for the true value. These two properties imply a parallel repetition bound as $$ \mathrm{val}(G^{\otimes k}) \approx \mathrm{val}_+(G^{\otimes k}) = \mathrm{val}_+(G)^{k} \approx \mathrm{val}(G)^{k}. $$ Using this framework, we can also give a short proof for the NP-hardness of Label-Cover$(1,δ)$ for all $δ>0$, starting from the basic PCP theorem. We prove the following new results: - A parallel repetition bound for projection games with small soundness. Previously, it was not known whether parallel repetition decreases the value of such games. This result implies stronger inapproximability bounds for Set-Cover and Label-Cover. - An improved bound for few parallel repetitions of projection games, showing that Raz's counterexample is tight even for a small number of repetitions. Our techniques also allow us to bound the value of the direct product of multiple games, namely, a bound on $\mathrm{val}(G_1\otimes ...\otimes G_k)$ for different projection games $G_1,...,G_k$.

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arXiv2014

Derandomized Parallel Repetition via Structured PCPs

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. 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.

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arXiv2010

Dense locally testable codes cannot have constant rate and distance

A q-query locally testable code (LTC) is an error correcting code that can be tested by a randomized algorithm that reads at most q symbols from the given word. An important question is whether there exist LTCs that have the ccc-property: constant relative rate, constant relative distance, and that can be tested with a constant number of queries. Such codes are sometimes referred to as "asymptotically good". We show that dense LTCs cannot be ccc. The density of a tester is roughly the average number of distinct local views in which a coordinate participates. An LTC is dense if it has a tester with density >> 1. More precisely, we show that a 3-query locally testable code with a tester of density >> 1 cannot be ccc. Moreover, we show that a q-query locally testable code (q>3) with a tester of density >> n^{q-2} cannot be ccc. Our results hold when the tester has the following two properties: 1) "no weights": Every q-tuple of queries occurs with the same probability. 2) "last-one-fixed": In every `test' of the tester, the value to any q-1 of the symbols determines the value of the last symbol. (Linear codes have constraints of this type). We also show that several natural ways to quantitatively improve our results would already resolve the general ccc-question, i.e. also for non-dense LTCs.

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arXiv2005

Conditional Hardness for Approximate Coloring

We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$ or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, 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 coloring is hard for any constant $\eps>0$, 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 $\eps$ fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least $\eps$. Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO'05].

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arXiv2003

A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

Given a $k$-uniform hyper-graph, the E$k$-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 E$k$-Vertex-Cover is NP-hard to approximate within factor $(k-1-ε)$ for any $k \geq 3$ 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.

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