arXiv subjects

Venkatesan Guruswami

Publications and source records attributed to Venkatesan Guruswami.

At least 19 recordsLinked to original sources

arXiv2026

Combinatorial limitations of list-decoding Reed-Solomon codes

For infinitely many rates $R \in (0,1)$, we show that full-length Reed-Solomon codes over fields of low characteristic can contain super-polynomially many codewords in a Hamming ball of radius just exceeding the Johnson radius. Over prime fields, we exhibit full-length Reed-Solomon codes of constant rate $R$ with polynomially many codewords in a Hamming ball of relative radius bounded strictly below $1-R$. This refutes a recent conjecture that, over prime fields, list sizes below capacity are always bounded by a constant. Together, these results demonstrate that certain restrictions in recent works on explicit Reed-Solomon codes achieving list-decoding capacity, related to the choice of field or evaluation points or to the list-size guarantee, are inherent.

cs.IT↗

arXiv2026

Upper Bounds for CSP Non-redundancy via Locally Decodable Codes

In the study of constraint satisfaction problems (CSPs), the non-redundancy of a relation measures the maximum number of clauses in a CSP instance which can be logically independent. In the last few years, CSP non-redundancy has found a number of applications including in query complexity, sparsification, kernelization and streaming. In this paper, we present a novel technique for upper-bounding the non-redundancy of some relations using locally decodable codes (LDCs) and locally correctable codes (LCCs). More precisely, known limitations on the structure of 2-query LDCs and LCCs can be translated into limitations on the size of certain non-redundant CSP instances. This technique builds on a technique used by Brakensiek and Guruswami [STOC 2025] to bound non-redundancy via techniques from matching vector codes. Using LDCs and LCCs, we give improved upper bounds on non-redundancy of the BCK predicate pioneered by Bessiere, Carbonnel, and Katsirelos [AAAI 2020] as well as its well-studied Boolean variant. We also give the first example of a Boolean predicate whose non-redundancy is provably not of the form $n^{k\pm o(1)}$ for some positive integer $k$.

cs.DM↗

arXiv2026

Improved LCL Equivalences and Curve Decoding Alphabet Sizes

Local Coordinate-wise Linear (LCL) properties, introduced by Levi, Mosheiff, and Shagrithaya (2025), capture coding-theoretic properties such as list decoding and list recovery. They showed that every LCL property has the same rate threshold for random linear codes (RLC) and random Reed--Solomon codes (RRS). Brakensiek, Chen, Dhar, and Zhang (2025) extended this to random additive codes. Thus any property of all subspace design codes holds for RLCs and RRS at the same rate. In this work, we improve the probabilistic guarantees in the RRS local equivalence theorem and extend the equivalence to all known subspace design constructions individually. Each such construction, including folded Reed--Solomon codes, has the same LCL threshold as RLC and RRS codes. We apply these results to alphabet-size bounds for curve decoding. Curve decoding, introduced by Goyal and Guruswami (2025) and Carmon et al. (2026), is a row-span-constrained LCL property implying proximity gaps (PG), correlated agreement (CA), and mutual correlated agreement (MCA), relevant to IOPs and SNARKs. Informally, a rate-$R$ code $C \subset \mathbb{F}^n$ is $(\ell,δ,a,b)$-curve decodable if any degree-$\ell$ curve $u:\mathbb{F}\to\mathbb{F}^n$ that is $δ$-close to $C$ on at least $a$ points has at least $b\leq a$ nearby codewords lying on a degree-$\ell$ curve in $C$. Goyal and Guruswami (2025) and Goyal, Guruswami, Sun, and Wootters (2026) established curve decodability for RLC and RRS codes at radius $δ$ approaching $1-R$, but with large alphabets in $\ell$ and $η:=1-R-δ$. Using our improved LCL equivalence and rank-sensitive counting inspired by Goyal and Guruswami (2026), we exponentially improve alphabet requirements while matching the best known values of $(a,b)$. For $\ell=O(1)$, our alphabet-size bounds match the best known for list decoding, and improves results for PG, CA, and MCA.

cs.IT↗

arXiv2026

Explicit Reed-Solomon Subspace Designs: Optimal List and Line Decoding

For any desired rate $R \in (0,1)$, and gap to list-decoding capacity $\varepsilon > 0$, we give an explicit $n^{O_{\varepsilon}(1)}$ time construction of Reed-Solomon codes over extension fields of size polynomial in the block length $n$ that can be list-decoded from a fraction $1-R-\varepsilon$ of errors with the optimal list-size of $O(\frac{1-R}{\varepsilon})$. We also give a simpler, strongly explicit construction of evaluation points for which the associated Reed-Solomon code can be list-decoded from $1-R-\varepsilon$ errors with list size $(1/\varepsilon)^{O(1/\varepsilon)}$. We provide a deterministic list-decoding algorithm for both constructions running in time $O_ε(1)\mathsf{poly}(n)$. In fact, our Reed-Solomon codes are near-optimal subspace-design codes over a subfield. Via known results, this directly implies the combinatorial list-decoding guarantee, as well as near-optimal list-recovery bounds. We also present a new reduction showing that the subspace-design guarantee also implies line decoding over the extension field itself, which then yields proximity gaps and mutual correlated agreement up to proximity parameter approaching the optimal $1-R$ radius. Compared to the recent algorithm of Brakensiek, Chen, Putterman, Zhang, and Zheng (2026), which list decodes every Reed-Solomon code over prime fields up to capacity, our approach achieves constant (in fact, near-optimal) list size rather than a polynomially growing one, at the cost of carefully choosing the code's evaluation points. In a companion paper, we show that these respective restrictions are unavoidable.

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arXiv2026

Average-Radius List-Decodability of Random Linear Codes

We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - ε$ is $(p, C_{p,q}/ε)$-average-radius list-decodable with probability at least $1 - q^{-Ω(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/ε$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, Håstad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/ε)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).

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arXiv2026

Tight Bounds for Sparsifying Random CSPs

The problem of CSP sparsification asks: for a given CSP instance, what is the sparsest possible reweighting such that for every possible assignment to the instance, the number of satisfied constraints is preserved up to a factor of $1 \pm ε$? We initiate the study of the sparsification of random CSPs. In particular, we consider two natural random models: the $r$-partite model and the uniform model. In the $r$-partite model, CSPs are formed by partitioning the variables into $r$ parts, with constraints selected by randomly picking one vertex out of each part. In the uniform model, $r$ distinct vertices are chosen at random from the pool of variables to form each constraint. In the $r$-partite model, we exhibit a sharp threshold phenomenon. For every predicate $P$, there is an integer $k$ such that a random instance on $n$ vertices and $m$ edges cannot (essentially) be sparsified if $m \le n^k$ and can be sparsified to size $\approx n^k$ if $m \ge n^k$. Here, $k$ corresponds to the largest copy of the AND which can be found within $P$. Furthermore, these sparsifiers are simple, as they can be constructed by i.i.d. sampling of the edges. In the uniform model, the situation is a bit more complex. For every predicate $P$, there is an integer $k$ such that a random instance on $n$ vertices and $m$ edges cannot (essentially) be sparsified if $m \le n^k$ and can sparsified to size $\approx n^k$ if $m \ge n^{k+1}$. However, for some predicates $P$, if $m \in [n^k, n^{k+1}]$, there may or may not be a nontrivial sparsifier. In fact, we show that there are predicates where the sparsifiability of random instances is non-monotone, i.e., as we add more random constraints, the instances become more sparsifiable. We give a precise (efficiently computable) procedure for determining which situation a specific predicate $P$ falls into.

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arXiv2026

Constant-Overhead Injection into Quantum Codes

We construct the first known fault-tolerant scheme for injecting states into quantum error-correcting codes with constant space and time overhead. That is, we construct a family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block. Similarly, a code state can be ejected, i.e. fault-tolerantly decoded, back into bare physical qubits. We show that these injection and ejection procedures succeed under circuit-level locally stochastic noise, while incurring just a small constant probability of corrupting each qubit, which is unavoidable for bare physical qubits. We also show how to perform fault-tolerant error correction and code-state preparation under locally stochastic noise. All of our gadgets can be implemented with constant quantum circuit depth (i.e. are single-shot), and with a number of physical qubits growing linearly with the number of logical qubits, assuming the ability to run polynomial-sized noiseless classical circuits on the side. We construct our quantum codes by taking a high-dimensional hypergraph product of classical LDPC codes, which in turn are a simplified version of Spielman's linear-time encodable codes (STOC'95). As our resulting product codes are only resilient to physical errors occurring with non-uniform probabilities across qubits, we then show how to concatenate with inner codes of various sizes to obtain fault-tolerance against uniform noise.

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arXiv2026

The Reach of Abelian Covers in Hypergraphs

Covers in hypergraphs are frequently studied to capture various forms of dependence between hyperedges. For example, even covers--which check if each vertex appears in an even number of hyperedges--have found much success recently in the study of locally decodable codes. Inspired by a recently-emerging line of work on the non-redundancy of constraint satisfaction problems (CSPs), we introduce and study two novel families of covers of hypergraphs which are stricter than even covers: \emph{Abelian} covers and Catalan covers. Abelian covers are similar to even covers, except that arithmetic is now done over the integers rather than modulo 2, allowing us to capture dependences over arbitrary Abelian groups. Catalan covers capture the behavior of non-Abelian groups by only allowing local cancellations in a sequence of hyperedges. We prove three main results about Abelian and Catalan covers. First, using tools from lattice theory, we show that any $r$-uniform hypergraph with $n$ vertices and $n \log(r)$ hyperedges has an Abelian cover. Second, using tools from algebraic topology, we show that in any $3$-uniform hypergraph, Abelian covers and Catalan covers are equivalent; thereby showing that Catalan covers emerge after $O(n)$ hyperedges in $3$-uniform hypergraphs. Finally, using the theory of nilpotent groups, we show that there exists a $4$-uniform hypergraph which has an Abelian cover but not a Catalan cover. Collectively, these results exactly characterize the reach that Abelian covers have in deducing dependences in hypergraphs. As our primary application, we show that any arity-$3$ CSP with an infinite-domain Mal'tsev extension has linear non-redundancy. This implies near optimal streaming, sparsification, and kernelization algorithms for this family of CSPs. Previously, such a result was only known for the much simpler case of arity-$2$ CSPs.

cs.DM↗

arXiv2026

The Richness of CSP Non-redundancy

In the field of constraint satisfaction problems (CSP), a clause is called redundant if its satisfaction is implied by satisfying all other clauses. An instance of CSP$(P)$ is called non-redundant if it does not contain any redundant clause. The non-redundancy (NRD) of a predicate $P$ is the maximum number of clauses in a non-redundant instance of CSP$(P)$, as a function of the number of variables $n$. Recent progress has shown that non-redundancy is crucially linked to many other important questions in computer science and mathematics including sparsification, kernelization, query complexity, universal algebra, and extremal combinatorics. Given that non-redundancy is a nexus for many of these important problems, the central goal of this paper is to more deeply understand non-redundancy. Our first main result shows that for every rational number $r \ge 1$, there exists a finite CSP predicate $P$ such that the non-redundancy of $P$ is $Θ(n^r)$. Our second main result explores the concept of conditional non-redundancy first coined by Brakensiek and Guruswami [STOC 2025]. We completely classify the conditional non-redundancy of all binary predicates (i.e., constraints on two variables) by connecting these non-redundancy problems to the structure of high-girth graphs in extremal combinatorics. Inspired by these concrete results, we build off the work of Carbonnel [CP 2022] to develop an algebraic theory of conditional non-redundancy. As an application of this algebraic theory, we revisit the notion of Mal'tsev embeddings, which is the most general technique known to date for establishing that a predicate has linear non-redundancy. For example, we provide the first example of predicate with a Mal'tsev embedding that cannot be attributed to the structure of an Abelian group, but rather to the structure of the quantum Pauli group.

cs.DM↗

arXiv2026

Super-linear Lower Bounds for CSP Non-Redundancy via Shrinking Instances

We say that an instance of a constraint satisfaction problem (CSP) is non-redundant if the satisfaction of each clause cannot be implied by the satisfaction of the other clauses in the instance. The non-redundancy (NRD) of a CSP is the maximal number of clauses a non-redundant instance can have for a given number of variables. NRD is closely tied to the behavior of CSPs in various computational models including their sparsification, kernelization, and streaming complexity. A primary open question in the study of non-redundancy is the identification of which CSP predicates have near-linear NRD. Recent works by Carbonnel [CP 2022], Khanna, Putterman and Sudan [STOC 2025], Brakensiek and Guruswami [STOC 2025] and Brakensiek, Guruswami, Jansen, Lagerkvist, and Wahlström [2025] have introduced various forms of gadget reductions between CSPs to relate their non-redundancy. The primary contribution of this work is to recontextualize many of these gadget reductions in a framework which we call hypergraph projections. By studying a quantity we call the shrinking factor of these hypergraph projections, we can more precisely predict when a gadget reduction between predicates can yield a super-linear NRD lower bound, greatly improving on the analysis of previous works. To illustrate the power of our framework, we identify some concrete CSP predicates whose non-redundancy is at the cusp of our understanding and show how our methods give lower bounds that could not have been achieved with previous methods. We also demonstrate how these gadget reductions can be automatically deduced using SAT solvers, thereby opening up novel computational avenues for discovering further relationships between the non-redundancy of various CSPs.

cs.DM↗

arXiv2026

Almost Optimal FPT Inapproximability for k-SetCover

We show that $\bigl(\frac{\log n}{\log\log n}\bigr)$-approximate parameterized $k$-SetCover is W[1]-hard, and has no $n^{o(k/\log k)}$-time algorithms under ETH. This improves upon the previous best factors $\bigl(\frac{\log n}{\log\log n}\bigr)^{1/k}$ in (Lin, 2019) and $(\log n)^{1/\operatorname{poly}(k)}$ in (Karthik, Laekhanukit, and Manurangsi, 2019). Here $k$ is the yes-case guarantee and $n$ is the number of candidate sets. While the best approximation ratio is still $O(\log n)$ via the greedy algorithm, closing this $1/k$ gap in the exponent has been a longstanding open problem; we remove this loss via a simple direct reduction. The construction is self-contained and does not rely on the parameterized inapproximability hypothesis (PIH). Starting with sparse parameterized 2-CSP instances (Karthik, Marx, Pilipczuk, and Souza, 2024), we build a monotone CNF formula, which is equivalent to a SetCover instance. To obtain a $k$-versus-$h$ gap, the reduction enumerates all hash functions from $Σ$ to $[2h]$ and all unsatisfiable 2-CSP instances on the same constraint graph with alphabet $[2h]$. For each such instance, it asks for a certificate that the hashed label pairs are not all contained in that instance. Perfect hashing makes this enumeration efficient for $h=\log n/\log\log n$.

cs.CC↗

arXiv2026

Separating Non-redundancy and Chain Length

For a constraint satisfaction problem defined by a relation $R$, its non-redundancy $\text{NRD}(R,n)$ is the size of largest instance (as a function of the number $n$ of variables) for which no constraint is implied by the rest. Its chain length $\text{CL}(R,n)$ is the largest such instance where the constraints can be ordered so that no constraint is implied by the preceding ones. Clearly $\text{CL}(R,n) \ge \text{NRD}(R,n)$ but so far no asymptotic separation was known between these quantities. We exhibit an explicit arity $4$ relation for which $\text{CL}(R,n) \ge ω(\text{NRD}(R,n))$.

cs.CC↗

arXiv2026

Binary code rate bounds via classical--quantum channels

We derive the four principal asymptotic rate-distance tradeoffs for binary codes---Plotkin, Elias--Bassalygo, and the two McEliece--Rodemich--Rumsey--Welch (MRRW) bounds---from one theorem, the ``pretty good criterion.'' If the bit error rate under the pretty good measurement (PGM)---the quantum analog of posterior sampling---of a binary-input output-symmetric classical--quantum (cq) channel lies below $δ$, then every length-$n$ binary code, linear or nonlinear, of relative distance $δ$ has rate at most the channel's capacity, up to an $O(n^{-1/2})$ correction. Rate--distance bounds thereby reduce to a channel design problem, wherein the task is to minimize channel capacity subject to the posterior bit error rate constraint. Via the pretty good criterion, the binary erasure channel (BEC) yields Plotkin, the binary symmetric channel (BSC) yields Elias--Bassalygo, the pure-state channel (PSC) yields the first MRRW bound, and a masked PSC yields the second MRRW bound exactly. This framework is then instantiated with new channels to improve upon the MRRW bounds. Specifically, the mixed-qubit channel (MQC), a mixed-state version of PSC, strictly improves the first MRRW bound at every $0 < δ< \frac{1}{2}$, while the masked mixed-qubit channel (2MQC) strictly improves the second MRRW bound throughout the same interval.

cs.IT↗

arXiv2026

Frequency Coding over Noisy Sampling

DNA molecules are so small that it might be practical to use their frequency vectors to encode messages. More precisely, a sender can inject $M_X$ copies of the string $X =$ CATCATCAT into a pool and the receiver can recover $M_X$ by sequencing the pool. There are, however, two sources of uncertainty: (a) $M_X$ is usually too big to be counted exactly, but is estimated by sampling. (b) The DNA sequencer could be noisy; it may have difficulty distinguishing CATCATCAT from CATGATCAT. Recently, Tamir, Weinberger, and Guillén i Fàbregas clarified the amount of information the frequency vector can carry under (a). They showed that each string can carry about $\log_4 R$ bits, where $R$ is the average number of times each string is read. They also showed that $\log_4 R$ bits can be achieved by a low-complexity uncoded scheme under the condition that there are at least $\sqrt R$ distinct strings. In this paper, we show that a low-complexity coded scheme can achieve the same $\log_4 R$ bits unconditionally. We then generalize the scheme to handle sequencing noise, (b), and show that the noise penalizes the total number of bits by $\log_2 \det W$, together with a linear term due to the use of Fourier transforms in our proof. The former penalty $\log_2 \det W$ is asymptotically the same as that obtained by Gerzon, Shomorony, and Weinberger; our scheme trades a small amount of rate for practical complexity.

cs.IT↗

arXiv2026

Inapproximability of Unique-Machine Precedence Scheduling for Unit-Length Jobs

The Unique-Machine Precedence Scheduling (UMPS) problem, introduced by [DKRSTZ22], seeks a makespan-minimizing schedule of precedence-constrained jobs when each job has a unique eligible machine. On the one hand, UMPS generalizes job shop scheduling by allowing the precedence graph to be an arbitrary DAG rather than a disjoint union of chains. On the other hand, UMPS admits approximation-preserving reductions to scheduling problems with communication delays, including the job-job delay model [DKRSTZ22] and the job-machine delay model [RSY23]. Despite its central role, the approximability of UMPS has remained poorly understood: even for unit-length jobs, known scheduling techniques do not seem to yield a non-trivial approximation, and the existence of a polylogarithmic approximation was left open by [DKRSTZ22]. On the hardness side, the previous best lower bound for unit-length jobs was only the 5/4 inherited from job shop scheduling [WHHHLSS97]. We prove that unit-length UMPS is NP-hard to approximate within any constant factor. We further show that, assuming NP is not in quasi-polynomial time, unit-length UMPS admits no polynomial-time $(\log n)^γ$-approximation for some constant $γ>0$. Via the known reductions from UMPS, these lower bounds also transfer to the corresponding unit-length communication-delay scheduling models. Our proof proceeds via a reduction from a hypergraph coloring promise problem. In the yes case, the input hypergraph admits a balanced coloring, while in the no case, the hypergraph has no large independent set. Instantiating this reduction with the hardness of [GL18] gives arbitrary constant-factor inapproximability, while combining the $4$-colorable $4$-uniform hypergraph coloring hardness of [GHHSV17] with a certain composition operation for hypergraphs yields the polylogarithmic factor inapproximability.

cs.DS↗

arXiv2026

Locality of Curve-Decoding and Improved Proximity Gaps

Proximity gaps are a property of error correcting codes that arise in the study of Interactive Oracle Proofs (IOPs) and Succinct Non-interactive Arguments of Zero Knowledge (SNARKs). Recent work of Goyal and Guruswami has established near-optimal proximity gaps for many families of codes, including subspace design codes, as well as random ensembles like random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble of LDPC codes (Goyal & Guruswami, 2025). However, the parameters for these latter randomized ensembles are worse than the parameters for subspace design codes, and degrade as the degree ell increases. In this work, we obtain improved proximity gaps for random ensembles of codes, including random linear codes, Reed-Solomon codes with random evaluation points, and Gallager's ensemble. Quantitatively, our results for these random ensembles match the results that Goyal and Guruswami attained for subspace design codes. In fact, our techniques are a black-box transference from subspace design codes: any progress on subspace design codes will automatically lead to analogous progress for these random ensembles. To obtain our results, we extend the Local Coordinate-wise Linear (LCL) property framework developed by Levi, Mosheiff, and Shagrithaya and by Brakensiek, Chen, Dhar, and Zhang to a \textit{row-span constrained} version (Levi, Mosheiff & Shagrithaya, 2025; Brakensiek, Chen, Dhar & Zhang, 2025). This allows us to cast \textit{curve-decodability} -- a property that implies proximity gaps -- directly as a row-span constrained LCL property, and make use of that machinery. In contrast, because curve-decodability is not obviously a vanilla LCL property, prior work had worked with a proxy property instead, leading to the aforementioned parameter losses.

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arXiv2026

Quantum Hierarchical Locally Recoverable Codes

Quantum locally recoverable codes (QLRCs) have recently gained attention as a framework for achieving efficient quantum storage with local recovery capabilities. Analogous to their classical counterparts, QLRCs allow a lost qudit to be reconstructed using only a small subset of other qudits, thereby reducing the resource and operational overhead in recovery. In this work, we extend the study of QLRCs by considering $(r,δ)$ QLRCs characterized by locality parameter $r$ and local distance $δ\geq 2$. We present constructions of both random and explicit $(r,δ)$ QLRCs, including explicit families based on the quantum Tamo--Barg construction. We also present an efficient decoding algorithm for these quantum Tamo--Barg codes. Furthermore, we introduce quantum \emph{hierarchical} locally recoverable codes (QHLRCs), which extend local recovery to multiple hierarchical levels. For any integer $h\geq 2$, we construct both random and explicit $h$-level QHLRCs, the latter being $h$-level quantum Tamo--Barg codes, and establish a Singleton-like bound for these codes using a CSS framework built from dual-containing classical codes. These results advance the theoretical foundations of quantum erasure recovery and contribute to the design of efficient quantum storage architectures.

cs.IT↗

arXiv2026

On the Approximability of Parameterized Minimum Monotone Satisfying Assignment

The parameterized Minimum Monotone Satisfying Assignment ($k$-MMSA) problem asks whether a monotone Boolean circuit admits a satisfying assignment of Hamming weight at most $k$. The MMSA hierarchy is defined by allowing a bounded number of alternations between AND and OR gates in the circuit. While the polynomial-time approximability of the MMSA hierarchy has been studied extensively, much less is known in the parameterized setting. In particular, $k$-MMSA$_2$ is the well-known $k$-SetCover problem, whose parameterized inapproximability lies in the $\text{polylog}(n)$ regime. In contrast, $k$-MMSA$_4$ captures $k$-MinLabel, for which known lower bounds give $\text{poly}(n)$ inapproximability. Sandwiched by $k$-MMSA$_2$ and $k$-MMSA$_4$, the inapproximability of $k$-MMSA$_3$ remained comparatively unexplored. In this paper, we give an FPT-time $O(2^k \log n)$-approximation algorithm for $k$-MMSA$_3$, suggesting that in the fixed-parameter regime, the third level of MMSA remains surprisingly close to the second level. Complementing this algorithm, we also give an FPT-time gap-preserving reduction from $k$-MMSA$_3$ to $k$-MMSA$_2$. Thus, stronger inapproximability for $k$-MMSA$_3$ would imply new hardness for $k$-MMSA$_2$, potentially offering a route around the current barriers for the latter problem. Revisiting Marx's reduction from $k$-MMSA$_t$ to gap $k$-MMSA$_{t+2}$, we also show that $k$-MMSA$_4$ admits no $n^{o(1)}$-factor FPT approximation unless W[2]=FPT, and no $n^{O(1/k)}$-factor approximation running in $n^{o(k)}$ time under ETH. These results separate the parameterized approximability behavior of the third and fourth levels and clarify where stronger inapproximability enters the $k$-MMSA hierarchy.

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