arXiv subjects

Venkatesan Guruswami

Publications and source records attributed to Venkatesan Guruswami.

At least 91 records · Page 5Linked to original sources

arXiv2022

Bypassing the XOR Trick: Stronger Certificates for Hypergraph Clique Number

Let $\mathcal{H}(k,n,p)$ be the distribution on $k$-uniform hypergraphs where every subset of $[n]$ of size $k$ is included as an hyperedge with probability $p$ independently. In this work, we design and analyze a simple spectral algorithm that certifies a bound on the size of the largest clique, $ω(H)$, in hypergraphs $H \sim \mathcal{H}(k,n,p)$. For example, for any constant $p$, with high probability over the choice of the hypergraph, our spectral algorithm certifies a bound of $\tilde{O}(\sqrt{n})$ on the clique number in polynomial time. This matches, up to $\textrm{polylog}(n)$ factors, the best known certificate for the clique number in random graphs, which is the special case of $k = 2$. Prior to our work, the best known refutation algorithms [CGL04, AOW15] rely on a reduction to the problem of refuting random $k$-XOR via Feige's XOR trick [Fei02], and yield a polynomially worse bound of $\tilde{O}(n^{3/4})$ on the clique number when $p = O(1)$. Our algorithm bypasses the XOR trick and relies instead on a natural generalization of the Lovasz theta semidefinite programming relaxation for cliques in hypergraphs.

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arXiv2022

$\ell_p$-Spread and Restricted Isometry Properties of Sparse Random Matrices

Random subspaces $X$ of $\mathbb{R}^n$ of dimension proportional to $n$ are, with high probability, well-spread with respect to the $\ell_2$-norm. Namely, every nonzero $x \in X$ is "robustly non-sparse" in the following sense: $x$ is $\varepsilon \|x\|_2$-far in $\ell_2$-distance from all $δn$-sparse vectors, for positive constants $\varepsilon, δ$ bounded away from $0$. This "$\ell_2$-spread" property is the natural counterpart, for subspaces over the reals, of the minimum distance of linear codes over finite fields, and corresponds to $X$ being a Euclidean section of the $\ell_1$ unit ball. Explicit $\ell_2$-spread subspaces of dimension $Ω(n)$, however, are unknown, and the best known constructions (which achieve weaker spread properties), are analogs of low density parity check (LDPC) codes over the reals, i.e., they are kernels of sparse matrices. We study the spread properties of the kernels of sparse random matrices. Rather surprisingly, we prove that with high probability such subspaces contain vectors $x$ that are $o(1)\cdot \|x\|_2$-close to $o(n)$-sparse with respect to the $\ell_2$-norm, and in particular are not $\ell_2$-spread. On the other hand, for $p < 2$ we prove that such subspaces are $\ell_p$-spread with high probability. Moreover, we show that a random sparse matrix has the stronger restricted isometry property (RIP) with respect to the $\ell_p$ norm, and this follows solely from the unique expansion of a random biregular graph, yielding a somewhat unexpected generalization of a similar result for the $\ell_1$ norm [BGI+08]. Instantiating this with explicit expanders, we obtain the first explicit constructions of $\ell_p$-RIP matrices for $1 \leq p < p_0$, where $1 < p_0 < 2$ is an absolute constant.

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arXiv2022

General Strong Polarization

Arikan's exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix $M$, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the {\em polarization} of an associated $[0,1]$-bounded martingale, namely its convergence in the limit to either $0$ or $1$. Arikan showed polarization of the martingale associated with the matrix $G_2 = \left(\begin{matrix} 1& 0 1& 1\end{matrix}\right)$ to get capacity achieving codes. His analysis was later extended to all matrices $M$ that satisfy an obvious necessary condition for polarization. While Arikan's theorem does not guarantee that the codes achieve capacity at small blocklengths, it turns out that a "strong" analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with $G_2$ such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE IT '14]), resolving a major theoretical challenge of the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are also simpler and modular. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving codes for arbitrary symmetric memoryless channels. We show how to use our analyses to achieve exponentially small error probabilities at lengths inverse polynomial in the gap to capacity. Indeed we show that we can essentially match any error probability with lengths that are only inverse polynomial in the gap to capacity.

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arXiv2022

Visible Rank and Codes with Locality

We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call \emph{visible rank}. The locality constraints of a linear code are stipulated by a matrix $H$ of $\star$'s and $0$'s (which we call a "stencil"), whose rows correspond to the local parity checks (with the $\star$'s indicating the support of the check). The visible rank of $H$ is the largest $r$ for which there is a $r \times r$ submatrix in $H$ with a unique generalized diagonal of $\star$'s. The visible rank yields a field-independent combinatorial lower bound on the rank of $H$ and thus the co-dimension of the code. We prove a rank-nullity type theorem relating visible rank to the rank of an associated construct called \emph{symmetric spanoid}, which was introduced by Dvir, Gopi, Gu, and Wigderson~\cite{DGGW20}. Using this connection and a construction of appropriate stencils, we answer a question posed in \cite{DGGW20} and demonstrate that symmetric spanoid rank cannot improve the currently best known $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-query locally correctable codes (LCCs) of length $n$. We also study the $t$-Disjoint Repair Group Property ($t$-DRGP) of codes where each codeword symbol must belong to $t$ disjoint check equations. It is known that linear $2$-DRGP codes must have co-dimension $Ω(\sqrt{n})$. We show that there are stencils corresponding to $2$-DRGP with visible rank as small as $O(\log n)$. However, we show the second tensor of any $2$-DRGP stencil has visible rank $Ω(n)$, thus recovering the $Ω(\sqrt{n})$ lower bound for $2$-DRGP. For $q$-LCC, however, the $k$'th tensor power for $k\le n^{o(1)}$ is unable to improve the $\widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-LCCs by a polynomial factor.

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arXiv2022

Arıkan meets Shannon: Polar codes with near-optimal convergence to channel capacity

Let $W$ be a binary-input memoryless symmetric (BMS) channel with Shannon capacity $I(W)$ and fix any $α> 0$. We construct, for any sufficiently small $δ> 0$, binary linear codes of block length $O(1/δ^{2+α})$ and rate $I(W)-δ$ that enable reliable communication on $W$ with quasi-linear time encoding and decoding. Shannon's noisy coding theorem established the \emph{existence} of such codes (without efficient constructions or decoding) with block length $O(1/δ^2)$. This quadratic dependence on the gap $δ$ to capacity is known to be best possible. Our result thus yields a constructive version of Shannon's theorem with near-optimal convergence to capacity as a function of the block length. This resolves a central theoretical challenge associated with the attainment of Shannon capacity. Previously such a result was only known for the erasure channel. Our codes are a variant of Arıkan's polar codes based on multiple carefully constructed local kernels, one for each intermediate channel that arises in the decoding. A crucial ingredient in the analysis is a strong converse of the noisy coding theorem when communicating using random linear codes on arbitrary BMS channels. Our converse theorem shows extreme unpredictability of even a single message bit for random coding at rates slightly above capacity.

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arXiv2021

Beyond Single-Deletion Correcting Codes: Substitutions and Transpositions

We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, or one substitution, with nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations: We construct linear-time encodable and decodable length-$n$ non-binary codes correcting a single edit error with nearly optimal redundancy $\log n+O(\log\log n)$, providing an alternative simpler proof of a result by Cai, Chee, Gabrys, Kiah, and Nguyen (IEEE Trans. Inf. Theory 2021). This is achieved by employing what we call weighted VT sketches, a notion that may be of independent interest. We construct linear-time encodable and list-decodable binary codes with list-size $2$ for one deletion and one substitution with redundancy $4\log n+O(\log\log n)$. This matches the existential bound up to an $O(\log\log n)$ additive term. We show the existence of a binary code correcting one deletion or one adjacent transposition with nearly optimal redundancy $\log n+O(\log\log n)$.

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arXiv2021

An Exponential Lower Bound on the Sub-Packetization of MSR Codes

An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a code is called minimum storage regenerating (MSR), if any single symbol of a codeword can be recovered by downloading $\ell/r$ field elements (which is known to be the least possible) from each of the other symbols. MSR codes are attractive for use in distributed storage systems, and by now a variety of ingenious constructions of MSR codes are available. However, they all suffer from exponentially large sub-packetization $\ell \gtrsim r^{k/r}$. Our main result is an almost tight lower bound showing that for an MSR code, one must have $\ell \ge \exp(Ω(k/r))$. This settles a central open question concerning MSR codes that has received much attention. Previously, a lower bound of $\approx \exp(\sqrt{k/r})$, and a tight lower bound for a restricted class of "optimal access" MSR codes, were known.

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arXiv2021

Linear Shannon Capacity of Cayley Graphs

The Shannon capacity of a graph is a fundamental quantity in zero-error information theory measuring the rate of growth of independent sets in graph powers. Despite being well-studied, this quantity continues to hold several mysteries. Lovász famously proved that the Shannon capacity of $C_5$ (the 5-cycle) is at most $\sqrt{5}$ via his theta function. This bound is achieved by a simple linear code over $\mathbb{F}_5$ mapping $x \mapsto 2x$. This motivates the notion of linear Shannon capacity of graphs, which is the largest rate achievable when restricting oneself to linear codes. We give a simple proof based on the polynomial method that the linear Shannon capacity of $C_5$ is $\sqrt{5}$. Our method applies more generally to Cayley graphs over the additive group of finite fields $\mathbb{F}_q$, giving an upper bound on the linear Shannon capacity. We compare this bound to the Lovász theta function, showing that they match for self-complementary Cayley graphs (such as $C_5$), and that the bound is smaller in some cases. We also exhibit a quadratic gap between linear and general Shannon capacity for some graphs.

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arXiv2021

CNF Satisfiability in a Subspace and Related Problems

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over $\F_2$ each of which is a product of affine forms. We focus on the case of k-CNF formulas (the k-SUB-SAT problem). Clearly, it is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show NP-hardness for k=2 and W[1]-hardness parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-2-SUB-SAT is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances. On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with runtime 1.5^r for 2-SUB-SAT, where $r$ is the subspace dimension and an O^*(1.4312)^n time algorithm where $n$ is the number of variables. For k more than 2, while known algorithms for solving a system of degree $k$ polynomial equations already imply a solution with runtime 2^{r(1-1/2k)}, we explore a more combinatorial approach. For instance, based on the notion of critical variables, we give an algorithm with running time ${n\choose {\le t}} 2^{n-n/k}$, where $n$ is the number of variables and $t$ is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our algorithm also achieves polynomial space in contrast to the algebraic approach that uses exponential space.

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arXiv2021

Linear Programming Bounds for Almost-Balanced Binary Codes

We revisit the linear programming bounds for the size vs. distance trade-off for binary codes, focusing on the bounds for the almost-balanced case, when all pairwise distances are between $d$ and $n-d$, where $d$ is the code distance and $n$ is the block length. We give an optimal solution to Delsarte's LP for the almost-balanced case with large distance $d \geq (n - \sqrt{n})/2 + 1$, which shows that the optimal value of the LP coincides with the Grey-Rankin bound for self-complementary codes. We also show that a limitation of the asymptotic LP bound shown by Samorodnitsky, namely that it is at least the average of the first MRRW upper bound and Gilbert-Varshamov bound, continues to hold for the almost-balanced case.

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arXiv2021

AC-DC: Amplification Curve Diagnostics for Covid-19 Group Testing

The first part of the paper presents a review of the gold-standard testing protocol for Covid-19, real-time, reverse transcriptase PCR, and its properties and associated measurement data such as amplification curves that can guide the development of appropriate and accurate adaptive group testing protocols. The second part of the paper is concerned with examining various off-the-shelf group testing methods for Covid-19 and identifying their strengths and weaknesses for the application at hand. The third part of the paper contains a collection of new analytical results for adaptive semiquantitative group testing with probabilistic and combinatorial priors, including performance bounds, algorithmic solutions, and noisy testing protocols. The probabilistic setting is of special importance as it is designed to be simple to implement by nonexperts and handle heavy hitters. The worst-case paradigm extends and improves upon prior work on semiquantitative group testing with and without specialized PCR noise models.

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arXiv2021

Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy

A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or $\mathsf{NP}$-hard. This paper considers a promise-problem variant of CSPs called PCSPs. A PCSP over a finite set of pairs of constraints $Γ$ consists of a pair $(Ψ_P, Ψ_Q)$ of CSPs with the same set of variables such that for every $(P, Q) \in Γ$, $P(x_{i_1}, ..., x_{i_k})$ is a clause of $Ψ_P$ if and only if $Q(x_{i_1}, ..., x_{i_k})$ is a clause of $Ψ_Q$. The promise problem $\operatorname{PCSP}(Γ)$ is to distinguish, given $(Ψ_P, Ψ_Q)$, between the cases $Ψ_P$ is satisfiable and $Ψ_Q$ is unsatisfiable. Many natural problems including approximate graph and hypergraph coloring can be placed in this framework. This paper is motivated by the pursuit of understanding the computational complexity of Boolean promise CSPs. As our main result, we show that $\operatorname{PCSP}(Γ)$ exhibits a dichotomy (it is either polynomial time solvable or $\mathsf{NP}$-hard) when the relations in $Γ$ are symmetric and allow for negations of variables. We achieve our dichotomy theorem by extending the weak polymorphism framework of Austrin, Guruswami, and Håstad [FOCS '14] which itself is a generalization of the algebraic approach to study CSPs. In both the algorithm and hardness portions of our proof, we incorporate new ideas and techniques not utilized in the CSP case. Furthermore, we show that the computational complexity of any promise CSP (over arbitrary finite domains) is captured entirely by its weak polymorphisms, a feature known as Galois correspondence, as well as give necessary and sufficient conditions for the structure of this set of weak polymorphisms. Such insights call us to question the existence of a general dichotomy for Boolean PCSPs.

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arXiv2020

Optimal rate list decoding over bounded alphabets using algebraic-geometric codes

We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction $1-R-ε$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $ε$. The alphabet size depends only $ε$ and is nearly-optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The list decoding algorithm is based on a linear-algebraic approach, which pins down the candidate messages to a subspace with a nice "periodic" structure. The list is pruned by precoding into a special form of "subspace-evasive" sets, which are constructed pseudorandomly. Instantiating this construction with the Garcia-Stichtenoth function field tower yields codes list-decodable up to a $1-R-ε$ error fraction with list size bounded by $O(1/ε)$, matching the existential bound up to constant factors. The parameters we achieve are thus quite close to the existential bounds in all three aspects: error-correction radius, alphabet size, and list-size. The second class of codes are obtained by restricting evaluation points of an algebraic-geometric code to rational points from a subfield. Once again, the linear-algebraic approach to list decoding to pin down candidate messages to a periodic subspace. We develop an alternate approach based on "subspace designs" to precode messages. Together with the subsequent explicit constructions of subspace designs, this yields a deterministic construction of an algebraic code family of rate $R$ with efficient list decoding from $1-R-ε$ fraction of errors over a constant-sized alphabet. The list size is bounded by a very slowly growing function of the block length $N$; in particular, it is at most $O(\log^{(r)} N)$ (the $r$'th iterated logarithm) for any fixed integer $r$.

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arXiv2020

Approximate Hypergraph Vertex Cover and generalized Tuza's conjecture

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this conjecture, and also studied its implied fractional versions. We establish the fractional version of the Aharoni-Zerbib conjecture up to lower order terms. Specifically, we give a factor $t/2+ O(\sqrt{t \log t})$ approximation based on LP rounding for an algorithmic version of the hypergraph Turán problem (AHTP). The objective in AHTP is to pick the smallest collection of $(t-1)$-sized subsets of vertices of an input $t$-uniform hypergraph such that every hyperedge contains one of these subsets. Aharoni and Zerbib also posed whether Tuza's conjecture and its hypergraph versions could follow from non-trivial duality gaps between vertex covers and matchings on hypergraphs that exclude certain sub-hypergraphs, for instance, a "tent" structure that cannot occur in the incidence of triangles and edges. We give a strong negative answer to this question, by exhibiting tent-free hypergraphs, and indeed $\mathcal{F}$-free hypergraphs for any finite family $\mathcal{F}$ of excluded sub-hypergraphs, whose vertex covers must include almost all the vertices. The algorithmic questions arising in the above study can be phrased as instances of vertex cover on simple hypergraphs, whose hyperedges can pairwise share at most one vertex. We prove that the trivial factor $t$ approximation for vertex cover is hard to improve for simple $t$-uniform hypergraphs. However, for set cover on simple $n$-vertex hypergraphs, the greedy algorithm achieves a factor $(\ln n)/2$, better than the optimal $\ln n$ factor for general hypergraphs.

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arXiv2020

The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs

In the field of constraint satisfaction problems (CSP), promise CSPs are an exciting new direction of study. In a promise CSP, each constraint comes in two forms: "strict" and "weak," and in the associated decision problem one must distinguish between being able to satisfy all the strict constraints versus not being able to satisfy all the weak constraints. The most commonly cited example of a promise CSP is the approximate graph coloring problem--which has recently seen exciting progress [BKO19, WZ20] benefiting from a systematic algebraic approach to promise CSPs based on "polymorphisms," operations that map tuples in the strict form of each constraint to tuples in the corresponding weak form. In this work, we present a simple algorithm which in polynomial time solves the decision problem for all promise CSPs that admit infinitely many symmetric polymorphisms, which are invariant under arbitrary coordinate permutations. This generalizes previous work of the first two authors [BG19]. We also extend this algorithm to a more general class of block-symmetric polymorphisms. As a corollary, this single algorithm solves all polynomial-time tractable Boolean CSPs simultaneously. These results give a new perspective on Schaefer's classic dichotomy theorem and shed further light on how symmetries of polymorphisms enable algorithms. Finally, we show that block symmetric polymorphisms are not only sufficient but also necessary for this algorithm to work, thus establishing its precise power

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arXiv2020

Strongly refuting all semi-random Boolean CSPs

We give an efficient algorithm to strongly refute \emph{semi-random} instances of all Boolean constraint satisfaction problems. The number of constraints required by our algorithm matches (up to polylogarithmic factors) the best-known bounds for efficient refutation of fully random instances. Our main technical contribution is an algorithm to strongly refute semi-random instances of the Boolean $k$-XOR problem on $n$ variables that have $\widetilde{O}(n^{k/2})$ constraints. (In a semi-random $k$-XOR instance, the equations can be arbitrary and only the right-hand sides are random.) One of our key insights is to identify a simple combinatorial property of random XOR instances that makes spectral refutation work. Our approach involves taking an instance that does not satisfy this property (i.e., is \emph{not} pseudorandom) and reducing it to a partitioned collection of $2$-XOR instances. We analyze these subinstances using a carefully chosen quadratic form as a proxy, which in turn is bounded via a combination of spectral methods and semidefinite programming. The analysis of our spectral bounds relies only on an off-the-shelf matrix Bernstein inequality. Even for the purely random case, this leads to a shorter proof compared to the ones in the literature that rely on problem-specific trace-moment computations.

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arXiv2020

Explicit two-deletion codes with redundancy matching the existential bound

We give an explicit construction of length-$n$ binary codes capable of correcting the deletion of two bits that have size $2^n/n^{4+o(1)}$. This matches up to lower order terms the existential result, based on an inefficient greedy choice of codewords, that guarantees such codes of size $Ω(2^n/n^4)$. Our construction is based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations. We also give an explicit construction of binary codes of size $Ω(2^n/n^{3+o(1)})$ that can be list decoded from two deletions using lists of size two. Previously, even the existence of such codes was not clear.

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arXiv2020

Bounds for list-decoding and list-recovery of random linear codes

A family of error-correcting codes is list-decodable from error fraction $p$ if, for every code in the family, the number of codewords in any Hamming ball of fractional radius $p$ is less than some integer $L$ that is independent of the code length. It is said to be list-recoverable for input list size $\ell$ if for every sufficiently large subset of codewords (of size $L$ or more), there is a coordinate where the codewords take more than $\ell$ values. The parameter $L$ is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size $L \to \infty$, is known to be $1-h_q(p)$ for list-decoding, and $1-\log_q \ell$ for list-recovery, where $q$ is the alphabet size of the code family. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below, $ε> 0$ is the gap to capacity). (1) A random linear code of rate $1 - \log_q(\ell) - ε$ requires list size $L \ge \ell^{Ω(1/ε)}$ for list-recovery from input list size $\ell$. This is surprisingly in contrast to completely random codes, where $L = O(\ell/ε)$ suffices w.h.p. (2) A random linear code of rate $1 - h_q(p) - ε$ requires list size $L \ge \lfloor h_q(p)/ε+0.99 \rfloor$ for list-decoding from error fraction $p$, when $ε$ is sufficiently small. (3) A random binary linear code of rate $1 - h_2(p) - ε$ is list-decodable from average error fraction $p$ with list size with $L \leq \lfloor h_2(p)/ε\rfloor + 2$. The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values.

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