arXiv subjects

Venkatesan Guruswami

Publications and source records attributed to Venkatesan Guruswami.

At least 37 records · Page 2Linked to original sources

arXiv2026

Explicit Constant-Alphabet Subspace Design Codes

The subspace design property for additive codes is a higher-dimensional generalization of the minimum distance property. As shown recently by Brakensiek, Chen, Dhar and Zhang, it implies that the code has similar performance as random linear codes with respect to all "local properties". Explicit algebraic codes, such as folded Reed-Solomon and multiplicity codes, are known to have the subspace design property, but they need alphabet sizes that grow as a large polynomial in the block length. Constructing explicit constant-alphabet subspace design codes was subsequently posed as an open question in Brakensiek, Chen, Dhar and Zhang. In this work, we answer their question and give explicit constructions of subspace design codes over constant-sized alphabets, using the expander-based Alon-Edmonds-Luby (AEL) framework. This generalizes the recent work of Jeronimo and Shagrithaya, which showed that such codes share local properties of random linear codes. Our work obtains this consequence in a unified manner via the subspace design property. In addition, our approach yields some improvements in parameters for list-recovery.

cs.IT↗

arXiv2026

Classification of Non-redundancy of Boolean Predicates of Arity 4

Given a constraint satisfaction problem (CSP) predicate $P \subseteq D^r$, the non-redundancy (NRD) of $P$ is maximum-sized instance on $n$ variables such that for every clause of the instance, there is an assignment which satisfies all but that clause. The study of NRD for various CSPs is an active area of research which combines ideas from extremal combinatorics, logic, lattice theory, and other techniques. Complete classifications are known in the cases $r=2$ and $(|D|=2, r=3)$. In this paper, we give a near-complete classification of the case $(|D|=2, r=4)$. Of the 400 distinct non-trivial Boolean predicates of arity 4, we implement an algorithmic procedure which perfectly classifies 397 of them. Of the remaining three, we solve two by reducing to extremal combinatorics problems -- leaving the last one as an open question. Along the way, we identify the first Boolean predicate whose non-redundancy asymptotics are non-polynomial.

cs.CC↗

arXiv2026

Maximal Recoverability: A Nexus of Coding Theory

In the modern era of large-scale computing systems, a crucial use of error correcting codes is to judiciously introduce redundancy to ensure recoverability from failure. To get the most out of every byte, practitioners and theorists have introduced the framework of maximal recoverability (MR) to study optimal error-correcting codes in various architectures. In this survey, we dive into the study of two families of MR codes: MR locally recoverable codes (LRCs) (also known as partial MDS codes) and grid codes (GCs). For each of these two families of codes, we discuss the primary recoverability guarantees as well as what is known concerning optimal constructions. Along the way, we discuss many surprising connections between MR codes and broader questions in computer science and mathematics. For MR LRCs, the use of skew polynomial codes has unified many previous constructions. For MR GCs, the theory of higher order MDS codes shows that MR GCs can be used to construct optimal list-decodable codes. Furthermore, the optimally recoverable patterns of MR GCs have close ties to long-standing problems on the structural rigidity of graphs.

cs.IT↗

arXiv2026

New Algorithms and Hardness Results for Robust Satisfiability of (Promise) CSPs

In this paper, we continue the study of robust satisfiability of promise CSPs (PCSPs), initiated in (Brakensiek, Guruswami, Sandeep, STOC 2023 / Discrete Analysis 2025), and obtain the following results: For the PCSP 1-in-3-SAT vs NAE-SAT with negations, we prove that it is hard, under the Unique Games conjecture (UGC), to satisfy $1-Ω(1/\log (1/ε))$ constraints in a $(1-ε)$-satisfiable instance. This shows that the exponential loss incurred by the BGS algorithm for the case of Alternating-Threshold polymorphisms is necessary, in contrast to the polynomial loss achievable for Majority polymorphisms. For any Boolean PCSP that admits Majority polymorphisms, we give an algorithm satisfying $1-O(\sqrtε)$ fraction of the weaker constraints when promised the existence of an assignment satisfying $1-ε$ fraction of the stronger constraints. This significantly generalizes the Charikar--Makarychev--Makarychev algorithm for 2-SAT, and matches the optimal trade-off possible under the UGC. The algorithm also extends, with the loss of an extra $\log (1/ε)$ factor, to PCSPs on larger domains with a certain structural condition, which is implied by, e.g., a family of Plurality polymorphisms. We prove that assuming the UGC, robust satisfiability is preserved under the addition of equality constraints. As a consequence, we can extend the rich algebraic techniques for decision/search PCSPs to robust PCSPs. The methods involve the development of a correlated and robust version of the general SDP rounding algorithm for CSPs due to (Brown-Cohen, Raghavendra, ICALP 2016), which might be of independent interest.

cs.DS↗

arXiv2025

Structure Theorems (and Fast Algorithms) for List Recovery of Subspace-Design Codes

List recovery of error-correcting codes has emerged as a fundamental notion with broad applications across coding theory and theoretical computer science. Folded Reed-Solomon (FRS) and univariate multiplicity codes are explicit constructions which can be efficiently list-recovered up to capacity, namely a fraction of errors approaching $1-R$ where $R$ is the code rate. Chen and Zhang and related works showed that folded Reed-Solomon codes and linear codes must have list sizes exponential in $1/ε$ for list-recovering from an error-fraction $1-R-ε$. These results suggest that one cannot list-recover FRS codes in time that is also polynomial in $1/ε$. In contrast to such limitations, we show, extending algorithmic advances of Ashvinkumar, Habib, and Srivastava for list decoding, that even if the lists in the case of list-recovery are large, they are highly structured. In particular, we can output a compact description of a set of size only $\ell^{O((\log \ell)/ε)}$ which contains the relevant list, while running in time only polynomial in $1/ε$ (the previously known compact description due to Guruswami and Wang had size $\approx n^{\ell/ε}$). We also improve on the state-of-the-art algorithmic results for the task of list-recovery.

cs.IT↗

arXiv2025

Block Length Gain for Nanopore Channels

DNA is an attractive candidate for data storage. Its millennial durability and nanometer scale offer exceptional data density and longevity. Its relevance to medical applications also drives advances in DNA-related biotechnology. To protect our data against errors, a straightforward approach uses one error-correcting code per DNA strand, with a Reed--Solomon code protecting the collection of strands. A downside is that current technology can only synthesize strands 200--300 nucleotides long. At this block length, the inner code rate suffers a significant finite-length penalty, making its effective capacity hard to characterize. Last year, we proposed $\textit{Geno-Weaving}$ in a JSAIT publication. The idea is to protect the same position across multiple strands using one code; this provably achieves capacity against substitution errors. In this paper, we extend the idea to combat deletion errors and show two more advantages of Geno-Weaving: (1) Because the number of strands is 3--4 orders of magnitude larger than the strand length, the finite-length penalty vanishes. (2) At realistic deletion rates $0.1\%$--$10\%$, Geno-Weaving designed for BSCs works well empirically, bypassing the need to tailor the design for deletion channels.

cs.IT↗

arXiv2025

Scheduling Problems with Constrained Rejections

We study bicriteria versions of Makespan Minimization on Unrelated Machines and Santa Claus by allowing a constrained number of rejections. Given an instance of Makespan Minimization on Unrelated Machines where the optimal makespan for scheduling $n$ jobs on $m$ unrelated machines is $T$, (Feige and Vondrák, 2006) gave an algorithm that schedules a $(1-1/e+10^{-180})$ fraction of jobs in time $T$. We show the ratio can be improved to $0.6533>1-1/e+0.02$ if we allow makespan $3T/2$. To the best our knowledge, this is the first result examining the tradeoff between makespan and the fraction of scheduled jobs when the makespan is not $T$ or $2T$. For the Santa Claus problem (the Max-Min version of Makespan Minimization), the analogous bicriteria objective was studied by (Golovin, 2005), who gave an algorithm providing an allocation so a $(1-1/k)$ fraction of agents receive value at least $T/k$, for any $k \in \mathbb{Z}^+$ and $T$ being the optimal minimum value every agent can receive. We provide the first hardness result by showing there are constants $δ,\varepsilon>0$ such that it is NP-hard to find an allocation where a $(1-δ)$ fraction of agents receive value at least $(1-\varepsilon) T$. To prove this hardness result, we introduce a bicriteria version of Set Packing, which may be of independent interest, and prove some algorithmic and hardness results for it. Overall, we believe these bicriteria scheduling problems warrant further study as they provide an interesting lens to understand how robust the difficulty of the original optimization goal might be.

cs.DS↗

arXiv2025

Near-Asymptotically-Good Quantum Codes with Transversal CCZ Gates and Sublinear-Weight Parity-Checks

It is a major challenge to construct good quantum codes supporting fault-tolerant (e.g. transversal) non-Clifford gates with low-weight parity-check measurements. In this paper, we construct the first known quantum codes with linear dimension and distance supporting transversal non-Clifford gates that have sublinear locality (i.e. parity-check weight). Specifically, we construct codes with transversal $CCZ$ gates that have dimension and distance $Θ(N)$ and locality $O(\sqrt{N})$, where $N$ denotes the block length. We furthermore design an efficient decoding algorithm for these codes. The alphabet size of these codes is $q=Θ(\sqrt{N})$, but it can be reduced to a constant (e.g. $q=2$) while incurring a polylogarithmic loss in other parameters. We also show how to decrease the locality to $O(N^{1/3})$, albeit with a larger alphabet size and slightly lower distance. We construct these codes as products of classical codes with appropriate algebraic structure. While our quantum codes are subsystem codes with non-commuting gauge operators, we show they nevertheless permit error correction from noisy syndrome measurements. As byproducts, we prove multiple technical results of independent interest. In particular, our efficient decoder can be viewed as a new multivariate generalization of Prony's method for reconstructing a function from partial access to its Fourier transform. Meanwhile, our distance analysis involves new connections to the classical study of maximally recoverable codes. Our results on product codes also resolve a conjecture of Bravyi & Hastings (2014) in the large-alphabet regime, by providing a new construction of quantum codes with dimension and distance $Θ(N)$ and locality $N^ε$ for arbitrary $ε>0$.

quant-ph↗

arXiv2025

On Inapproximability of Reconfiguration Problems: PSPACE-Hardness and some Tight NP-Hardness Results

Recently, Ohsaka [STACS'23] put forth the Reconfiguration Inapproximability Hypothesis (RIH), which roughly asserts that there is some $ε>0$ such that given as input a $k$-CSP instance (for some constant $k$) over some constant sized alphabet, and two satisfying assignments $ψ_s$ and $ψ_t$, it is PSPACE-hard to find a sequence of assignments starting from $ψ_s$ and ending at $ψ_t$ such that every assignment in the sequence satisfies at least $(1-ε)$ fraction of the constraints and also that every assignment in the sequence is obtained by changing its immediately preceding assignment (in the sequence) on exactly one variable. Assuming RIH, many important reconfiguration problems have been shown to be PSPACE-hard to approximate by Ohsaka [STACS'23; SODA'24]. In this paper, we provide a proof of RIH. Our proof uses known constructions of PCP of Proximity to create the gap, and further leverages a parallelization framework from recent parameterized inapproximability results to analyze the quantitative trade-off between $ε$ and $k$ in RIH. We note that Hirahara and Ohsaka [STOC'24] have also independently proved RIH. We also prove that the aforementioned $k$-CSP Reconfiguration problem is NP-hard to approximate to within a factor of $1/2 + ε$ (for any $ε>0$) when $k=2$. We complement this with a polynomial time $(1/2 - ε)$-approximation algorithm, which improves upon a $(1/4 - ε)$-approximation algorithm of Ohsaka [2023] (again for any $ε>0$). Finally, we show that Set Cover Reconfiguration is NP-hard to approximate to within a factor of $2 - ε$ for any constant $ε> 0$, which matches the simple linear-time 2-approximation algorithm by Ito et al. [TCS'11].

cs.CC↗

arXiv2025

SDPs and Robust Satisfiability of Promise CSP

For a constraint satisfaction problem (CSP), a robust satisfaction algorithm is one that outputs an assignment satisfying most of the constraints on instances that are near-satisfiable. It is known that the CSPs that admit efficient robust satisfaction algorithms are precisely those of bounded width, i.e., CSPs whose satisfiability can be checked by a simple local consistency algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact satisfiability of a bounded width CSP can be checked by combinatorial algorithms, the robust algorithm is based on rounding a canonical Semidefinite Programming (SDP) relaxation. In this work, we initiate the study of robust satisfaction algorithms for promise CSPs, which are a vast generalization of CSPs that have received much attention recently. The motivation is to extend the theory beyond CSPs, as well as to better understand the power of SDPs. We present robust SDP rounding algorithms under some general conditions, namely the existence of particular high-dimensional Boolean symmetries known as majority or alternating threshold polymorphisms. On the hardness front, we prove that the lack of such polymorphisms makes the PCSP hard for all pairs of symmetric Boolean predicates. Our approach relies on SDP integrality gaps argued via the absence of certain colorings of the sphere, with connections to sphere Ramsey theory. We conjecture that PCSPs with robust satisfaction algorithms are precisely those for which the feasibility of the canonical SDP implies (exact) satisfiability. We also give a precise algebraic condition, known as a minion characterization, of which PCSPs have the latter property.

cs.DS↗

arXiv2025

Random Reed-Solomon Codes Achieve List-Decoding Capacity With Linear-Sized Alphabets

Reed-Solomon codes are a classic family of error-correcting codes consisting of evaluations of low-degree polynomials over a finite field on some sequence of distinct field elements. They are widely known for their optimal unique-decoding capabilities, but their list-decoding capabilities are not fully understood. Given the prevalence of Reed-Solomon codes, a fundamental question in coding theory is determining if Reed-Solomon codes can optimally achieve list-decoding capacity. A recent breakthrough by Brakensiek, Gopi, and Makam established that Reed-Solomon codes are combinatorially list-decodable all the way to capacity. However, their results hold for randomly-punctured Reed-Solomon codes over an exponentially large field size $2^{O(n)}$, where $n$ is the block length of the code. A natural question is whether Reed-Solomon codes can still achieve capacity over smaller fields. We show that Reed-Solomon codes are list-decodable to capacity with linear field size $O(n)$, which is evidently optimal up to a constant factor. Our techniques also show that random linear codes are list-decodable up to capacity with optimal list-size $O(1/\varepsilon)$ and near-optimal alphabet size $2^{O(1/\varepsilon^2)}$, where $\varepsilon$ is the gap to capacity. As far as we are aware, list-decoding up to capacity with optimal list-size $O(1/\varepsilon)$ was not known to be achievable with any linear code over a constant alphabet size (even non-constructively), and it was also not known to be achievable for random linear codes over any alphabet size. With our proof, which maintains a hypergraph perspective of the list-decoding problem, we include an alternate presentation of ideas from Brakensiek, Gopi, and Makam that more directly connects the list-decoding problem to the GM-MDS theorem via a hypergraph orientation theorem.

cs.IT↗

arXiv2025

Cell-Probe Lower Bounds via Semi-Random CSP Refutation: Simplified and the Odd-Locality Case

A recent work (Korten, Pitassi, and Impagliazzo, FOCS 2025) established an insightful connection between static data structure lower bounds, range avoidance of $\text{NC}^0$ circuits, and the refutation of pseudorandom CSP instances, leading to improvements to some longstanding lower bounds in the cell-probe/bit-probe models. Here, we improve these lower bounds in certain cases via a more streamlined reduction to XOR refutation, coupled with handling the odd-arity case. Our result can be viewed as a complete derandomization of the state-of-the-art semi-random $k$-XOR refutation analysis (Guruswami, Kothari and Manohar, STOC 2022, Hsieh, Kothari and Mohanty, SODA 2023), which complements the derandomization of the even-arity case obtained by Korten et al. As our main technical statement, we show that for any multi-output constant-depth circuit that substantially stretches its input, its output is very likely far from strings sampled from distributions with sufficient independence, and further this can be efficiently certified. Via suitable shifts in perspectives, this gives applications to cell-probe lower bounds and range avoidance algorithms for $\mathsf{NC}^0$ circuits.

cs.CC↗

arXiv2025

New constructions of pseudorandom codes

Introduced in [CG24], pseudorandom error-correcting codes (PRCs) are a new cryptographic primitive with applications in watermarking generative AI models. These are codes where a collection of polynomially many codewords is computationally indistinguishable from random for an adversary that does not have the secret key, but anyone with the secret key is able to efficiently decode corrupted codewords. In this work, we examine the assumptions under which PRCs with robustness to a constant error rate exist. 1. We show that if both the planted hyperloop assumption introduced in [BKR23] and security of a version of Goldreich's PRG hold, then there exist public-key PRCs for which no efficient adversary can distinguish a polynomial number of codewords from random with better than $o(1)$ advantage. 2. We revisit the construction of [CG24] and show that it can be based on a wider range of assumptions than presented in [CG24]. To do this, we introduce a weakened version of the planted XOR assumption which we call the weak planted XOR assumption and which may be of independent interest. 3. We initiate the study of PRCs which are secure against space-bounded adversaries. We show how to construct secret-key PRCs of length $O(n)$ which are $\textit{unconditionally}$ indistinguishable from random by $\text{poly}(n)$ time, $O(n^{1.5-\varepsilon})$ space adversaries.

cs.CR↗

arXiv2025

PCPP-Based Reconfiguration Inapproximability: Query Complexity vs. Soundness Gap Trade-offs

The Reconfiguration Inapproximability Hypothesis (RIH), recently established by Hirahara-Ohsaka (STOC'24) and Karthik-Manurangsi (ECCC'24), studies the hardness of reconfiguring one solution into another in constraint satisfaction problems (CSP) when restricted to approximate intermediate solutions. In this work, we make a tighter connection between RIH's soundness gap and that of probabilistically checkable proofs of proximity (PCPP). Consequently, we achieve an improved trade-off between soundness and query complexity in Gap CSP Reconfiguration. Our approach leverages a parallelization framework, which also appears in some recent parameterized inapproximability results.

cs.CC↗

arXiv2025

Inapproximability of Finding Sparse Vectors in Codes, Subspaces, and Lattices

Finding sparse vectors is a fundamental problem that arises in several contexts including codes, subspaces, and lattices. In this work, we prove strong inapproximability results for all these variants using a novel approach that even bypasses the PCP theorem. Our main result is that it is NP-hard (under randomized reductions) to approximate the sparsest vector in a real subspace within any constant factor; the gap can be further amplified using tensoring. Our reduction has the property that there is a Boolean solution in the completeness case. As a corollary, this immediately recovers the state-of-the-art inapproximability factors for the shortest vector problem (SVP) on lattices. Our proof extends the range of $\ell_p$ (quasi) norms for which hardness was previously known, from $p\geq 1$ to all $p\geq 0$, answering a question raised by [Khot05]. Previous hardness results for SVP, and the related minimum distance problem (MDP) for error-correcting codes, all use lattice/coding gadgets that have an abundance of codewords in a ball of radius smaller than the minimum distance. In contrast, our reduction only needs many codewords in a ball of radius slightly larger than the minimum distance. This enables an easy derandomization of our reduction for finite fields, giving a new elementary proof of deterministic hardness for MDP. We believe this weaker density requirement might offer a promising approach to showing deterministic hardness of SVP, a long elusive goal. The key technical ingredient underlying our result for real subspaces is a proof that in the kernel of a random Rademacher matrix, the support of any two linearly independent vectors have very little overlap. A broader motivation behind this work is the development of inapproximability techniques for problems over the reals. We hope that the approach we develop could enable progress on analytic variants of sparsest vector.

cs.CC↗

arXiv2025

Semirandom Planted Clique via 1-norm Isometry Property

We give a polynomial-time algorithm that finds a planted clique of size $k \ge \sqrt{n \log n}$ in the semirandom model, improving the state-of-the-art $\sqrt{n} (\log n)^2$ bound. This $\textit{semirandom planted clique problem}$ concerns finding the planted subset $S$ of $k$ vertices of a graph $G$ on $V$, where the induced subgraph $G[S]$ is complete, the cut edges in $G[S; V \setminus S]$ are random, and the remaining edges in $G[V \setminus S]$ are adversarial. An elegant greedy algorithm by Blasiok, Buhai, Kothari, and Steurer [BBK24] finds $S$ by sampling inner products of the columns of the adjacency matrix of $G$, and checking if they deviate significantly from typical inner products of random vectors. Their analysis uses a suitably random matrix that, with high probability, satisfies a certain restricted isometry property. Inspired by Wootters's work on list decoding, we put forth and implement the $1$-norm analog of this argument, and quantitatively improve their analysis to work all the way up to the conjectured optimal $\sqrt{n \log n}$ bound on clique size, answering one of the main open questions posed in [BBK24].

cs.DS↗

arXiv2025

PCP-free APX-Hardness of Nearest Codeword and Minimum Distance

We give simple deterministic reductions demonstrating the NP-hardness of approximating the nearest codeword problem and minimum distance problem within arbitrary constant factors (and almost-polynomial factors assuming NP cannot be solved in quasipolynomial time). The starting point is a simple NP-hardness result without a gap, and is thus "PCP-free." Our approach is inspired by that of Bhattiprolu and Lee [BL24] who give a PCP-free randomized reduction for similar problems over the integers and the reals. We leverage the existence of $\varepsilon$-balanced codes to derandomize and further simplify their reduction for the case of finite fields.

cs.CC↗

arXiv2025

A Deterministic Construction of a Large Distance Code from the Wozencraft Ensemble

We present an explicit construction of a sequence of rate $1/2$ Wozencraft ensemble codes (over any fixed finite field $\mathbb{F}_q$) that achieve minimum distance $Ω(\sqrt{k})$ where $k$ is the message length. The coefficients of the Wozencraft ensemble codes are constructed using Sidon Sets and the cyclic structure of $\mathbb{F}_{q^{k}}$ where $k+1$ is prime with $q$ a primitive root modulo $k+1$. Assuming Artin's conjecture, there are infinitely many such $k$ for any prime power $q$.

cs.IT↗