arXiv subjects

Venkatesan Guruswami

Publications and source records attributed to Venkatesan Guruswami.

At least 109 records · Page 6Linked to original sources

arXiv2020

Optimally Resilient Codes for List-Decoding from Insertions and Deletions

We give a complete answer to the following basic question: "What is the maximal fraction of deletions or insertions tolerable by $q$-ary list-decodable codes with non-vanishing information rate?" This question has been open even for binary codes, including the restriction to the binary insertion-only setting, where the best-known result was that a $γ\leq 0.707$ fraction of insertions is tolerable by some binary code family. For any desired $ε> 0$, we construct a family of binary codes of positive rate which can be efficiently list-decoded from any combination of $γ$ fraction of insertions and $δ$ fraction of deletions as long as $ γ+2δ\leq 1-ε$. On the other hand, for any $γ,δ$ with $γ+2δ=1$ list-decoding is impossible. Our result thus precisely characterizes the feasibility region of binary list-decodable codes for insertions and deletions. We further generalize our result to codes over any finite alphabet of size $q$. Surprisingly, our work reveals that the feasibility region for $q>2$ is not the natural generalization of the binary bound above. We provide tight upper and lower bounds that precisely pin down the feasibility region, which turns out to have a $(q-1)$-piece-wise linear boundary whose $q$ corner-points lie on a quadratic curve. The main technical work in our results is proving the existence of code families of sufficiently large size with good list-decoding properties for any combination of $δ,γ$ within the claimed feasibility region. We achieve this via an intricate analysis of codes introduced by [Bukh, Ma; SIAM J. Discrete Math; 2014]. Finally, we give a simple yet powerful concatenation scheme for list-decodable insertion-deletion codes which transforms any such (non-efficient) code family (with vanishing information rate) into an efficiently decodable code family with constant rate.

cs.IT↗

arXiv2020

A locality-based approach for coded computation

Modern distributed computation infrastructures are often plagued by unavailabilities such as failing or slow servers. These unavailabilities adversely affect the tail latency of computation in distributed infrastructures. The simple solution of replicating computation entails significant resource overhead. Coded computation has emerged as a resource-efficient alternative, wherein multiple units of data are encoded to create parity units and the function to be computed is applied to each of these units on distinct servers. A decoder can use the available function outputs to decode the unavailable ones. Existing coded computation approaches are resource efficient only for simple variants of linear functions such as multilinear, with even the class of low degree polynomials requiring the same multiplicative overhead as replication for practically relevant straggler tolerance. In this paper, we present a new approach to model coded computation via the lens of locality of codes. We introduce a generalized notion of locality, denoted computational locality, building upon the locality of an appropriately defined code. We show that computational locality is equivalent to the required number of workers for coded computation and leverage results from the well-studied locality of codes to design coded computation schemes. We show that recent results on coded computation of multivariate polynomials can be derived using local recovering schemes for Reed-Muller codes. We present coded computation schemes for multivariate polynomials that adaptively exploit locality properties of input data-- an inadmissible technique under existing frameworks. These schemes require fewer workers than the lower bound under existing coded computation frameworks, showing that the existing multiplicative overhead on the number of servers is not fundamental for coded computation of nonlinear functions.

cs.IT↗

arXiv2019

Approximating Operator Norms via Generalized Krivine Rounding

We consider the $(\ell_p,\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A$ over vectors $x,y$ with $\|x\|_p=\|y\|_r=1$. The problem is equivalent to computing the $p \to r^*$ operator norm of $A$. The case $p=r=\infty$ corresponds to the classical Grothendieck problem. Our main result is an algorithm for arbitrary $p,r \ge 2$ with approximation ratio $(1+ε_0)/(\sinh^{-1}(1)\cdot γ_{p^*} \,γ_{r^*})$ for some fixed $ε_0 \le 0.00863$. Comparing this with Krivine's approximation ratio of $(π/2)/\sinh^{-1}(1)$ for the original Grothendieck problem, our guarantee is off from the best known hardness factor of $(γ_{p^*} γ_{r^*})^{-1}$ for the problem by a factor similar to Krivine's defect. Our approximation follows by bounding the value of the natural vector relaxation for the problem which is convex when $p,r \ge 2$. We give a generalization of random hyperplane rounding and relate the performance of this rounding to certain hypergeometric functions, which prescribe necessary transformations to the vector solution before the rounding is applied. Unlike Krivine's Rounding where the relevant hypergeometric function was $\arcsin$, we have to study a family of hypergeometric functions. The bulk of our technical work then involves methods from complex analysis to gain detailed information about the Taylor series coefficients of the inverses of these hypergeometric functions, which then dictate our approximation factor. Our result also implies improved bounds for "factorization through $\ell_{2}^{\,n}$" of operators from $\ell_{p}^{\,n}$ to $\ell_{q}^{\,m}$ (when $p\geq 2 \geq q$)--- such bounds are of significant interest in functional analysis and our work provides modest supplementary evidence for an intriguing parallel between factorizability, and constant-factor approximability.

cs.DS↗

arXiv2019

Near-optimal Repair of Reed-Solomon Codes with Low Sub-packetization

Minimum storage regenerating (MSR) codes are MDS codes which allow for recovery of any single erased symbol with optimal repair bandwidth, based on the smallest possible fraction of the contents downloaded from each of the other symbols. Recently, certain Reed-Solomon codes were constructed which are MSR. However, the sub-packetization of these codes is exponentially large, growing like $n^{Ω(n)}$ in the constant-rate regime. In this work, we study the relaxed notion of $ε$-MSR codes, which incur a factor of $(1+ε)$ higher than the optimal repair bandwidth, in the context of Reed-Solomon codes. We give constructions of constant-rate $ε$-MSR Reed-Solomon codes with polynomial sub-packetization of $n^{O(1/ε)}$ and thereby giving an explicit tradeoff between the repair bandwidth and sub-packetization.

cs.DS↗

arXiv2019

Leakage-Resilient Non-Malleable Secret Sharing in Non-compartmentalized Models

Non-malleable secret sharing was recently proposed by Goyal and Kumar in independent tampering and joint tampering models for threshold secret sharing (STOC18) and secret sharing with general access structure (CRYPTO18). The idea of making secret sharing non-malleable received great attention and by now has generated many papers exploring new frontiers in this topic, such as multiple-time tampering and adding leakage resiliency to the one-shot tampering model. Non-compartmentalized tampering model was first studied by Agrawal et.al (CRYPTO15) for non-malleability against permutation composed with bit-wise independent tampering, and shown useful in constructing non-malleable string commitments. We initiate the study of leakage-resilient secret sharing in the non-compartmentalized model. The leakage adversary can corrupt several players and obtain their shares, as in normal secret sharing. The leakage adversary can apply arbitrary affine functions with bounded total output length to the full share vector and obtain the outputs as leakage. These two processes can be both non-adaptive and do not depend on each other, or both adaptive and depend on each other with arbitrary ordering. We construct such leakage-resilient secret sharing schemes and achieve constant information ratio (the scheme for non-adaptive adversary is near optimal). We then explore making the non-compartmentalized leakage-resilient secret sharing also non-malleable against tampering. We consider a tampering model, where the adversary can use the shares obtained from the corrupted players and the outputs of the global leakage functions to choose a tampering function from a tampering family F. We give two constructions of such leakage-resilient non-malleable secret sharing for the case F is the bit-wise independent tampering and, respectively, for the case F is the affine tampering functions.

cs.CR↗

arXiv2019

Polynomial time decodable codes for the binary deletion channel

In the random deletion channel, each bit is deleted independently with probability $p$. For the random deletion channel, the existence of codes of rate $(1-p)/9$, and thus bounded away from $0$ for any $p < 1$, has been known. We give an explicit construction with polynomial time encoding and deletion correction algorithms with rate $c_0 (1-p)$ for an absolute constant $c_0 > 0$.

cs.IT↗

arXiv2019

Efficient Low-Redundancy Codes for Correcting Multiple Deletions

We consider the problem of constructing binary codes to recover from $k$-bit deletions with efficient encoding/decoding, for a fixed $k$. The single deletion case is well understood, with the Varshamov-Tenengolts-Levenshtein code from 1965 giving an asymptotically optimal construction with $\approx 2^n/n$ codewords of length $n$, i.e., at most $\log n$ bits of redundancy. However, even for the case of two deletions, there was no known explicit construction with redundancy less than $n^{Ω(1)}$. For any fixed $k$, we construct a binary code with $c_k \log n$ redundancy that can be decoded from $k$ deletions in $O_k(n \log^4 n)$ time. The coefficient $c_k$ can be taken to be $O(k^2 \log k)$, which is only quadratically worse than the optimal, non-constructive bound of $O(k)$. We also indicate how to modify this code to allow for a combination of up to $k$ insertions and deletions.

cs.IT↗

arXiv2019

Parameterized Inapproximability of Exact Cover and Nearest Codeword

The $k$-ExactCover problem is a parameterized version of the ExactCover problem, in which we are given a universe $U$, a collection $S$ of subsets of $U$, and an integer $k$, and the task is to determine whether $U$ can be partitioned into $k$ sets in $S$. This is a natural extension of the well-studied SetCover problem; though in the parameterized regime we know it to be $W[1]$-complete in the exact case, its parameterized complexity with respect to approximability is not well understood. We prove that, assuming ETH, for some $γ> 0$ there is no time $f(k) \cdot N^{γk}$ algorithm that can, given a $k$-ExactCover instance $I$, distinguish between the case where $I$ has an exact cover of size $k$ and the case where every set cover of $I$ has size at least $\frac14 \sqrt[k]{\frac{\log N}{\log \log N}}$. This rules out even more than FPT algorithms, and additionally rules out any algorithm whose approximation ratio depends only on the parameter $k$. By assuming SETH, we instead improve the lower bound to requiring time $f(k) \cdot N^{k - \varepsilon}$, for any $\varepsilon > 0$. In this work we also extend the inapproximability result to the $k$-Nearest-Codeword ($k$-NCP) problem. Specifically, given a generator matrix $A \in \mathbb{F}_2^{m \times n}$, a vector $y \in \mathbb{F}_2^m$, and the parameter $k$, we show that it is hard to distinguish between the case where there exists a codeword with distance at most $k$ from $y$ and the case where every codeword has distance at least $\frac18 \sqrt[k]{\frac{\log N}{\log \log N}}$ from $y$. This improves the best known parameterized inapproximability result, which rules out approximations with a factor of $\text{poly} (\log k)$, but requires us to assume ETH instead of $W[1] \neq FPT$.

cs.CC↗

arXiv2019

Bridging between 0/1 and Linear Programming via Random Walks

Under the Strong Exponential Time Hypothesis, an integer linear program with $n$ Boolean-valued variables and $m$ equations cannot be solved in $c^n$ time for any constant $c < 2$. If the domain of the variables is relaxed to $[0,1]$, the associated linear program can of course be solved in polynomial time. In this work, we give a natural algorithmic bridging between these extremes of $0$-$1$ and linear programming. Specifically, for any subset (finite union of intervals) $E \subset [0,1]$ containing $\{0,1\}$, we give a random-walk based algorithm with runtime $O_E((2-\text{measure}(E))^n\text{poly}(n,m))$ that finds a solution in $E^n$ to any $n$-variable linear program with $m$ constraints that is feasible over $\{0,1\}^n$. Note that as $E$ expands from $\{0,1\}$ to $[0,1]$, the runtime improves smoothly from $2^n$ to polynomial. Taking $E = [0,1/k) \cup (1-1/k,1]$ in our result yields as a corollary a randomized $(2-2/k)^{n}\text{poly}(n)$ time algorithm for $k$-SAT. While our approach has some high level resemblance to Schöning's beautiful algorithm, our general algorithm is based on a more sophisticated random walk that incorporates several new ingredients, such as a multiplicative potential to measure progress, a judicious choice of starting distribution, and a time varying distribution for the evolution of the random walk that is itself computed via an LP at each step (a solution to which is guaranteed based on the minimax theorem). Plugging the LP algorithm into our earlier polymorphic framework yields fast exponential algorithms for any CSP (like $k$-SAT, $1$-in-$3$-SAT, NAE $k$-SAT) that admit so-called `threshold partial polymorphisms.'

cs.DS↗

arXiv2019

Explicit optimal-length locally repairable codes of distance 5

Locally repairable codes (LRCs) have received significant recent attention as a method of designing data storage systems robust to server failure. Optimal LRCs offer the ideal trade-off between minimum distance and locality, a measure of the cost of repairing a single codeword symbol. For optimal LRCs with minimum distance greater than or equal to 5, block length is bounded by a polynomial function of alphabet size. In this paper, we give explicit constructions of optimal-length (in terms of alphabet size), optimal LRCs with minimum distance equal to 5.

cs.IT↗

arXiv2019

CSPs with Global Modular Constraints: Algorithms and Hardness via Polynomial Representations

We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo M, for various choices of the modulus M. Due to the known classification of tractable Boolean CSPs, this mainly reduces to the study of three cases: 2-SAT, HORN-SAT, and LIN-2 (linear equations mod 2). We classify the moduli M for which these respective problems are polynomial time solvable, and when they are not (assuming the ETH). Our study reveals that this modular constraint lends a surprising richness to these classic, well-studied problems, with interesting broader connections to complexity theory and coding theory. The HORN-SAT case is connected to the covering complexity of polynomials representing the NAND function mod M. The LIN-2 case is tied to the sparsity of polynomials representing the OR function mod M, which in turn has connections to modular weight distribution properties of linear codes and locally decodable codes. In both cases, the analysis of our algorithm as well as the hardness reduction rely on these polynomial representations, highlighting an interesting algebraic common ground between hard cases for our algorithms and the gadgets which show hardness. These new complexity measures of polynomial representations merit further study. The inspiration for our study comes from a recent work by Nägele, Sudakov, and Zenklusen on submodular minimization with a global congruence constraint. Our algorithm for HORN-SAT has strong similarities to their algorithm, and in particular identical kind of set systems arise in both cases. Our connection to polynomial representations leads to a simpler analysis of such set systems, and also sheds light on (but does not resolve) the complexity of submodular minimization with a congruency requirement modulo a composite M.

cs.CC↗

arXiv2018

Secret Sharing with Binary Shares

Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length $\ell$ among any $N \leq 2^\ell$ players such that for a threshold parameter $t$, (i) the knowledge of any $t$ shares does not reveal any information about the secret and, (ii) any choice of $t+1$ shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length $\ell$, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any $t+g$ shares, for a positive integer gap parameter $g$. Ramp secret sharing scheme necessarily requires shares of length $\ell/g$. Other than the bound related to secret length $\ell$, the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio $g/N$. In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio $g/N$, where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any $τ$ fraction of observed shares, and reconstruction from any $ρ$ fraction of shares, for any choices of $0 \leq τ< ρ\leq 1$. Our construction achieves secret length $N(ρ-τ-o(1))$, which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length $Ω(N(ρ-τ))$.

cs.CR↗

arXiv2018

Maximally Recoverable LRCs: A field size lower bound and constructions for few heavy parities

The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. These codes can correct a small number of erasures by accessing only a small number of remaining coordinates. An $(n,r,h,a,q)$-LRC is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $g=n/r$ local groups each of size $r$. It has $h$ global parity checks and each local group has $a$ local parity checks. Such an LRC is Maximally Recoverable (MR), if it corrects all erasure patterns which are information-theoretically correctable under the stipulated structure of local and global parity checks. We show the first non-trivial lower bounds on the field size required for MR LRCs. When $a,h$ are constant and the number of local groups $g \ge h$, while $r$ may grow with $n$, our lower bound simplifies to $q\ge Ω_{a,h}\left(n\cdot r^{\min\{a,h-2\}}\right).$ No superlinear (in $n$) lower bounds were known prior to this work for any setting of parameters. MR LRCs deployed in practice have a small number of global parities, typically $h=2,3$. We complement our lower bounds by giving constructions with small field size for $h\le 3$. For $h=2$, we give a linear field size construction. We also show a surprising application of elliptic curves and arithmetic progression free sets in the construction of MR LRCs.

cs.IT↗

arXiv2018

Streaming Hardness of Unique Games

We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by $p$, the alphabet size of the Unique Game, gives a trivial $p$-approximation that can be computed in $O(\log n)$ space. Meanwhile, with high probability, a sample of $\tilde{O}(n)$ constraints suffices to estimate the optimal value to $(1+ε)$ accuracy. We prove that any single-pass streaming algorithm that achieves a $(p-ε)$-approximation requires $Ω_ε(\sqrt{n})$ space. Our proof is via a reduction from lower bounds for a communication problem that is a $p$-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to down\emph{stream} hardness results for streaming approximability for other CSP-like problems.

cs.CC↗

arXiv2018

Polar Codes with exponentially small error at finite block length

We show that the entire class of polar codes (up to a natural necessary condition) converge to capacity at block lengths polynomial in the gap to capacity, while simultaneously achieving failure probabilities that are exponentially small in the block length (i.e., decoding fails with probability $\exp(-N^{Ω(1)})$ for codes of length $N$). Previously this combination was known only for one specific family within the class of polar codes, whereas we establish this whenever the polar code exhibits a condition necessary for any polarization. Our results adapt and strengthen a local analysis of polar codes due to the authors with Nakkiran and Rudra [Proc. STOC 2018]. Their analysis related the time-local behavior of a martingale to its global convergence, and this allowed them to prove that the broad class of polar codes converge to capacity at polynomial block lengths. Their analysis easily adapts to show exponentially small failure probabilities, provided the associated martingale, the ``Arikan martingale'', exhibits a corresponding strong local effect. The main contribution of this work is a much stronger local analysis of the Arikan martingale. This leads to the general result claimed above. In addition to our general result, we also show, for the first time, polar codes that achieve failure probability $\exp(-N^β)$ for any $β< 1$ while converging to capacity at block length polynomial in the gap to capacity. Finally we also show that the ``local'' approach can be combined with any analysis of failure probability of an arbitrary polar code to get essentially the same failure probability while achieving block length polynomial in the gap to capacity.

cs.IT↗

arXiv2018

Algorithmic Polarization for Hidden Markov Models

Using a mild variant of polar codes we design linear compression schemes compressing Hidden Markov sources (where the source is a Markov chain, but whose state is not necessarily observable from its output), and to decode from Hidden Markov channels (where the channel has a state and the error introduced depends on the state). We give the first polynomial time algorithms that manage to compress and decompress (or encode and decode) at input lengths that are polynomial $\it{both}$ in the gap to capacity and the mixing time of the Markov chain. Prior work achieved capacity only asymptotically in the limit of large lengths, and polynomial bounds were not available with respect to either the gap to capacity or mixing time. Our results operate in the setting where the source (or the channel) is $\it{known}$. If the source is $\it{unknown}$ then compression at such short lengths would lead to effective algorithms for learning parity with noise -- thus our results are the first to suggest a separation between the complexity of the problem when the source is known versus when it is unknown.

cs.IT↗

arXiv2018

Constructions of maximally recoverable local reconstruction codes via function fields

Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an $(n,r,h,a)$-LRC, the $n$ codeword symbols are partitioned into $r$ disjoint groups each of which include $a$ local parity checks capable of locally correcting $a$ erasures. MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size that exponential in $h$ or $a$, and it is of interest to obtain MR LRCs of minimal possible field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. For instance, for the case of small $r \ll ε\log n$ and large $h \ge Ω(n^{1-ε})$, we improve the field size from roughly $n^h$ to $n^{εh}$. For the case of $a=1$ (one local parity check), we improve the field size quadratically from $r^{h(h+1)}$ to $r^{h \lfloor (h+1)/2 \rfloor}$ for some range of $r$. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea.

cs.IT↗

arXiv2018

Inapproximability of Matrix $p\rightarrow q$ Norms

We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\, R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$, $q=1$), and has been widely studied in various regimes. When $p \geq q$, the problem exhibits a dichotomy: constant factor approximation algorithms are known if $2 \in [q,p]$, and the problem is hard to approximate within almost polynomial factors when $2 \notin [q,p]$. The regime when $p < q$, known as \emph{hypercontractive norms}, is particularly significant for various applications but much less well understood. The case with $p = 2$ and $q > 2$ was studied by [Barak et al, STOC'12] who gave sub-exponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the Exponential Time Hypothesis. However, no NP-hardness of approximation is known for these problems for any $p < q$. We study the hardness of approximating matrix norms in both the above cases and prove the following results: - We show that for any $1< p < q < \infty$ with $2 \notin [p,q]$, $\|A\|_{p\rightarrow q}$ is hard to approximate within $2^{O(\log^{1-ε}\!n)}$ assuming $NP \not\subseteq BPTIME(2^{\log^{O(1)}\!n})$. This suggests that, similar to the case of $p \geq q$, the hypercontractive setting may be qualitatively different when $2$ does not lie between $p$ and $q$. - For all $p \geq q$ with $2 \in [q,p]$, we show $\|A\|_{p\rightarrow q}$ is hard to approximate within any factor than $1/(γ_{p^*} \cdot γ_q)$, where for any $r$, $γ_r$ denotes the $r^{th}$ norm of a gaussian, and $p^*$ is the dual norm of $p$.

cs.CC↗