On Approximability of Satisfiable $k$-CSPs: VI

Amey Bhangale; Dor Minzer; Subhash Khot; Yang P. Liu

REVIEW 3 major objections 4 minor 1 cited by

On Approximability of Satisfiable $k$-CSPs: VI

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read These inverse theorems assert that any three bounded functions with noticeable 3-wise correlation must locally resemble product functions, and globally resemble a low-degree function times a product function.

desk verdict Important paper with a real gap in Lemma 5.4; deserves refereeing but not acceptance as is. read the letter →

arxiv 2411.15133 v1 pith:5PRMZBXF submitted 2024-11-22 cs.CC math.CO

Amey Bhangale , Subhash Khot , Yang P. Liu , Dor Minzer This is my paper · ORCID

classification cs.CCmath.CO MSC 11B3005D4068Q17

keywords inversetheorems3-wisecorrelationsswapnormpairwise-connecteddistributionsproductfunctionsrestrictedarithmeticprogressionsdirectsumtestingdensityincrement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is trying to establish a structural inverse theorem for three-way correlations on product spaces. It claims that when a pairwise-connected distribution $\mu$ over $\Sigma \times \Gamma \times \Phi$ has atom probabilities at least $\alpha$, and three 1-bounded functions $f,g,h$ have $|\mathbb{E}_{(x,y,z)\sim\mu^{\otimes n}}[f(x)g(y)h(z)]| \geq \varepsilon$, the correlation cannot be a rare accident: a random restriction of $f$ (and likewise $g,h$) down to about $\delta n$ coordinates must be $\delta$-correlated with a product function, with $\delta = \exp(-\varepsilon^{-O_\alpha(1)})$. The same structure upgrades to a global statement, where $f$ is correlated with a low-degree function times a product function. The paper shows these theorems are strong enough to give the first bounds with only three iterated logarithms for the restricted 3-AP problem over finite fields, and to analyze the diamond direct-sum test in the low-soundness regime.

What carries the argument

The central object is the swap norm, defined by $\mathrm{swap}(f) = \mathbb{E}_{x,y\sim\Sigma^n,\,(x',y')\sim(x\leftrightarrow y)}[f(x)f(y)f(x')f(y')]$, where for each coordinate the pair $(x'_i,y'_i)$ is either $(x_i,y_i)$ or $(y_i,x_i)$ independently. The paper proves that $\mathrm{swap}(f)^{1/4}$ is a norm and that it behaves like a $U^2$ Gowers norm adapted to product functions: product functions have swap norm 1, and a noticeable swap norm forces local correlation with a product function. The proof chain uses 'path tricks' to pass from 3-wise correlation to large swap norm, an induction on $\varepsilon$ that alternates random restrictions with singular-value decompositions to increase the swap norm, and finally a restriction inverse theorem built on a direct-product test and small-set expansion to convert local correlation into the global low-degree-times-product form.

What would settle it

Find a 1-bounded $f$ and $\varepsilon$ for which the set of product functions $P$ with $|\langle f,P\rangle| \geq \varepsilon$ cannot be covered by $1/\varepsilon^{2-\delta}$ product functions each within $\delta$ correlation of some short-list element, contradicting Lemma 7.5; a concrete candidate would be a function with exponentially many near-orthogonal product functions all $\varepsilon$-correlated with it. Alternatively, exhibit a pairwise-connected $\mu$ and functions $f,g,h$ with correlation at least $\varepsilon$ whose random restrictions never achieve the stated $\delta$-correlation with a product function at the claimed $\delta = \exp(-\varepsilon^{-O_\alpha(1)})$.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the only robust source of 3-wise correlation over pairwise-connected distributions is product structure, captured quantitatively by a local inverse theorem (Theorem 2) and a global inverse theorem (Theorem 1). The local statement says that with probability at least $\delta$, a random restriction of $f$ to $\delta n$ coordinates has correlation at least $\delta$ with a product function $\prod_{i\in I} P_i(x_i)$, where each $|P_i| \leq 1$. The global statement says that $f$ itself has correlation at least $\exp(-\exp((1/\varepsilon)^{O_\alpha(1)}))$ with $L \cdot P$, where $L$ has degree $\exp((1/\varepsilon)^{O_\alpha(1)})$, $\|L\|_2 \leq 1$, and $P$ is a product of single-coordinate phases. The paper's applications---the density bound for restricted 3-APs and the low-soundness diamond test analysis---are consequences of combining these inverse theorems with a density increment or with the direct-sum structure.

Load-bearing premise

The global upgrade rests on a lemma imported without proof from a companion paper: every 1-bounded function $f$ must have a short list of product functions such that every product function $\varepsilon$-correlated with $f$ is within $\delta$ correlation of some list element; if that lemma fails in the quantitative range the paper needs, the restriction-to-global theorem collapses.

Editorial extensions

If this is right

For every pairwise-connected distribution $\mu$ with minimum atom mass $\alpha$, any correlation $\geq \varepsilon$ forces random restrictions below $\delta n$ coordinates to exhibit product-function correlation with parameters tied to $\varepsilon$; this is the local inverse theorem (Theorem 2).
The global inverse theorem (Theorem 1) shows that the whole function correlates with $L \cdot P$, where $L$ has degree $\exp((1/\varepsilon)^{O_\alpha(1)})$ and the correlation is at least $\exp(-\exp((1/\varepsilon)^{O_\alpha(1)}))$; the low-degree factor $L$ is necessary, as the paper notes.
For the restricted 3-AP problem, any set $A \subseteq \Sigma^n$ with density at least $C_\Sigma/(\log \log \log n)^{c_\Sigma}$ contains a nontrivial triple whose coordinate-wise pattern lies in $S$, giving the first reasonable bounds over $\mathbb{F}_p$.
The diamond direct-sum test is analysed in the small-soundness regime: acceptance probability $1/p+\varepsilon$ implies correlation with $\omega_p^{\alpha f} L P$, with $L$ of degree $\exp((1/\varepsilon)^{O(1)})$ and $P$ a product function.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the imported short-list lemma (Lemma 7.5) holds in the full quantitative range, the same local-to-global template may extend to $k$-wise correlations and to distributions admitting $(\mathbb{Z},+)$ embeddings, where product functions play the role of characters.
The paper notes that both inverse theorems might hold with $\delta = \varepsilon^{O_\alpha(1)}$; if that improvement is real, the restricted 3-AP density bound would likely improve from triple-logarithmic to a substantially stronger form.
The swap norm is a general-purpose analytic tool for settings where the natural 'structured' objects are product functions rather than linear characters; it could be applied to other testing problems where the accepted family is a direct sum rather than a linear subspace.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. First, if the main theorem goes through, it is a real step forward for the satisfiable k-CSP program: it eliminates the no-(Z,+)-embedding barrier that bounded every previous paper in this series, and the swap norm is a genuinely useful new tool. Second, as written, the proof of the central local inverse theorem has a gap in Lemma 5.4 that keeps the induction from closing. The stress-test is right.

What is actually new and good: the swap norm is developed carefully; the path-trick argument in Section 3 is elegant; and the restricted 3-AP bound with triple-log density is a solid improvement over prior results. The paper is mostly self-contained and the high-level architecture—local swap-norm inverse theorem, then restriction inverse theorem, then applications—is clear.

The soft spot that matters is Lemma 5.4. The proof removes only the SVD terms with λ_i ≥ cε^4 and needs to show the remainder has small swap norm. It bounds swap(Δ−Δt)^{1/4} by the sum of the small singular values and then asserts that sum is ≤ cε^4. That assertion is not backed by anything in the setup: with t = O(ε^{-16}) terms, the ℓ1 sum of the small λ_i can be as large as ε^{-8} by Cauchy-Schwarz, and the termination condition on swap(Δt) does not control it. Without swap(Δ) small, the inequality swap(F)^{1/4} ≥ swap(f)^{1/4}(1−O(cε^2)) does not follow, so Case 2 collapses. Since Theorem 5 is the engine for Theorem 2, this is a load-bearing gap. It might be repairable by truncating the SVD at a threshold rather than at the top t terms, but the current text does not do that.

Smaller issues: Theorem 10's 'small perturbation' to ensure P(x)≠0 is unquantified; a referee should ask for the correlation loss. Lemma 7.5 is imported without proof from [BKM24a] and the direct-product-test argument depends on it; that is acceptable in a series but the quantitative form should be verified. Lemma 8.2's proof is sketchy around the probability–density tradeoff, though that looks fixable.

Who this is for: researchers in inverse theorems for low-degree correlations, CSP approximability, and additive combinatorics. It deserves peer review—the ideas are significant and the applications are real—but I would not accept it before Lemma 5.4 is fixed or replaced. Send it to a referee who will actually check the algebra, not desk reject.

Referee Report

3 major / 4 minor

Summary. The paper proves local and global inverse theorems for 3-wise correlations over pairwise-connected distributions, introducing a new 'swap norm' and using it to show that functions with nontrivial correlation must correlate, after random restriction, with product functions, and globally with a low-degree function times a product function. It derives applications to restricted 3-APs over finite fields and to direct sum testing in the low-soundness regime. The main technical work is the development of the swap norm, the local inverse theorem (Theorem 5) via an iterative SVD-and-restriction scheme, and the restriction-to-global upgrade (Theorem 9) via a direct product test.

Significance. If the results are correct, they resolve the main analytical question for approximating satisfiable 3-CSPs over pairwise-connected distributions, including distributions admitting (Z,+)-embeddings, and they give the first reasonable bounds for the restricted 3-AP problem over finite fields. The swap norm is a new and potentially useful tool, and the paper carefully reproduces the path-trick lemma and develops the restriction machinery from first principles. The applications to direct sum testing and to density increment arguments are substantive. However, the proof contains at least one quantitatively incorrect bound in the core induction and one unquantified perturbation step, so the central claims are not yet fully established as written.

major comments (3)

[Section 5.1.3, Lemma 5.4] The displayed chain bounding swap(Δ)^{1/4} is quantitatively wrong. The paper claims swap(Δ)^{1/4} ≤ swap(Δ_t)^{1/4} + swap(Δ−Δ_t)^{1/4} ≤ cε^4 + ∑_{i∈[t], λ_i<cε^4} λ_i ≤ 2cε^4, but the available hypotheses only give λ_i < cε^4 and t = O(ε^{-16}), so ∑ λ_i can be as large as t·cε^4 = cε^{-12}. The termination condition swap(Δ_t)^{1/2} ≤ c^2ε^8 does not control the ℓ1 sum of these singular values. This invalidates the claim that swap(Δ) is small, which is load-bearing for inequality (12) and for the subsequent Case 2 construction of g,h with increased swap norm. The argument may be repairable by bounding swap(Δ−Δ_t)^{1/4} by the ℓ2 norm (∑λ_i^2)^{1/2} and then using swap(Δ_t) to control ∑λ_i^4, but this repair is not present in the manuscript and must be supplied.
[Section 6, proof of Theorem 10] The proof begins 'By making small perturbations in P we may assume that P(x) ≠ 0 for all x', but no quantitative statement is given about how the correlation δ is affected. Since P_i are only assumed ℓ2-bounded, a factor P_i may vanish on a large set, and the correlation could in principle be concentrated on that zero set; a naive perturbation could destroy it. A rigorous replacement is needed, either by handling zeros directly or by proving a perturbation lemma with an explicit loss in δ.
[Section 7.2, use of Lemma 7.5] Lemma 7.5 is imported from [BKM24a] without proof, and it is indeed load-bearing for the restriction-to-global upgrade: it provides the short list ShortList_{ε,δ}[f] used to define W_I, ~W_I, and hence the direct product test F. Since the quantitative form (response to the short list size and the net property) is essential, the paper should either prove the lemma or state precisely the exact statement used here and verify that the quantitative parameters match the application. This is a dependency concern rather than an internal error, but it should be addressed for the paper to be self-contained at the level claimed.

minor comments (4)

[Section 7.1, Lemma 7.5] There is a typo in the statement: 'wuth' should be 'with'.
[Section 5.1.3] The notation 'swap(∆)^{1/4} ≤ cε^4 + ∑ λ_i' is unclear because the sum is over singular values of f, not of Δ_t; please clarify which terms are included and index the sum explicitly.
[Section 5.1, Lemma 5.2] In the proof, the event where Lemma 5.1 applies is stated with probability γ^2ε^5‖f_t‖_2^2/(8B^2), but later lower bounds occasionally replace this by γ^2ε^5; the constants should be tracked consistently.
[Section 8.1, proof of Theorem 3] The statement 'we may assume that item 2 of Lemma 8.2 holds, or else we have already obtained a density increment' is slightly informal; it would be clearer to explicitly split into the two cases and state the resulting invariant at each iteration.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central inverse theorems are derived internally; the only same-author import (Lemma 7.5) is an independent published lemma.

full rationale

I walked the derivation chain. Theorem 2 is assembled in-text from Theorem 7 (path-trick and symmetry argument, proved in Section 3.3), Theorem 5 (swap-norm inverse theorem, proved by induction in Section 5 with Lemmas 5.1–5.4), and Theorem 10 (bounded product conversion, proved in Section 6). Theorem 5's induction and Lemma 5.4 use SVD and previously proved properties of the swap norm; no parameter is fitted to the conclusion, and no conclusion is assumed in its own proof. Theorem 9's proof is carried out in Sections 7.2–7.4 using Theorem 11 and Theorem 12, both proved in-text. The only external load-bearing ingredient is Lemma 7.5, quoted from [BKM24a, Lemma 12.16]. This is a same-author citation, but it is a published, parameter-free structural lemma about short lists of product functions; its assumptions (a 1-bounded f and correlation thresholds ε, δ) do not include the conclusion of Theorem 9 or Theorem 1, so under the review rules it counts as independent prior evidence rather than circularity. The skeptic's concern about Lemma 5.4's discarded-singular-value bound is a possible quantitative correctness gap, not a circular step: the asserted bound may fail, but no equation is being reintroduced as its own input. No data, fitted parameters, or renamed empirical patterns enter the proof.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper is a pure mathematics result with no empirical fitting. The main inputs are the pairwise-connected distribution class with a uniform lower bound on atom probabilities, standard analytic inequalities, and one imported short-list lemma from prior work by the same authors. Constants such as γ, δ, and ζ are chosen within proofs but are not data-fitted parameters.

assumptions (4)

domain assumption Pairwise-connectedness of supp(μ_xy), supp(μ_xz), supp(μ_yz) and atom lower bound α
Theorems 1, 2, and 3 assume this class; Theorem 7's path-trick steps require connected supports to enlarge the distribution and make other supports full.
domain assumption ShortList lemma (Lemma 7.5) from [BKM24a, Lemma 12.16]
Imported without proof; used in Section 7.2 to construct short lists of product functions correlating with random restrictions, a key step in Theorem 9.
standard math Small-set expansion spectral bound in the biased hypercube (Theorem 13)
Used in Section 7.3 to prove Theorem 11; stated as a standard consequence of noise operator hypercontractivity from O'Donnell.
standard math Finite-dimensional SVD and the Eckart-Young optimal rank-t approximation property
Central tool in Sections 2, 4, and 5; the proofs assume that subtracting the top t singular terms optimally reduces the ℓ2 norm of the residual matrix.

invented entities (1)

Swap norm (swap(f)^{1/4})
purpose: Measures how much a function correlates with product functions under random coordinate swaps; it is the main new analytic device in Theorems 5 and 7.
The swap norm is well-defined and is proved to be a norm in Corollary 3.6, but it is an internal mathematical tool with no independent empirical or external confirmation beyond the theorems proved in this paper.

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Cite this review

Pith. "Pith review of On Approximability of Satisfiable $k$-CSPs: VI." pith.science (2026). https://pith.science/paper/5PRMZBXF

@misc{pith2026241115133,
  author       = {Pith},
  title        = {Pith review of: On Approximability of Satisfiable $k$-CSPs: VI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5PRMZBXF}},
  note         = {Machine review of arXiv:2411.15133}
}

abstract

We prove local and global inverse theorems for general $3$-wise correlations over pairwise-connected distributions. Let $\mu$ be a distribution over $\Sigma \times \Gamma \times \Phi$ such that the supports of $\mu_{xy}$, $\mu_{xz}$, and $\mu_{yz}$ are all connected, and let $f: \Sigma^n \to \mathbb{C}$, $g: \Gamma^n \to \mathbb{C}$, $h: \Phi^n \to \mathbb{C}$ be $1$-bounded functions satisfying \[ \left|\mathbb{E}_{(x,y,z) \sim \mu^{\otimes n}}[f(x)g(y)h(z)]\right| \geq \varepsilon. \] In this setting, our local inverse theorem asserts that there is $\delta :=\textsf{exp}(-\varepsilon^{-O_{\mu}(1)})$ such that with probability at least $\delta$, a random restriction of $f$ down to $\delta n$ coordinates $\delta$-correlates to a product function. To get a global inverse theorem, we prove a restriction inverse theorem for general product functions, stating that if a random restriction of $f$ down to $\delta n$ coordinates is $\delta$-correlated with a product function with probability at least $\delta$, then $f$ is $2^{-\textsf{poly}(\log(1/\delta))}$-correlated with a function of the form $L\cdot P$, where $L$ is a function of degree $\textsf{poly}(1/\delta)$, $\|L\|_2\leq 1$, and $P$ is a product function. We show applications to property testing and to additive combinatorics. In particular, we show the following result via a density increment argument. Let $\Sigma$ be a finite set and $S \subseteq \Sigma \times \Sigma \times \Sigma$ such that: (1) $(x, x, x) \in S$ for all $x \in S$, and (2) the supports of $S_{xy}$, $S_{xz}$, and $S_{yz}$ are all connected. Then, any set $A \subseteq \Sigma^n$ with $|\Sigma|^{-n}|A| \geq \Omega((\log \log \log n)^{-c})$ contains $x, y, z \in A$, not all equal, such that $(x_i,y_i,z_i) \in S$ for all $i$. This gives the first reasonable bounds for the restricted 3-AP problem over finite fields.

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Reference graph

Works this paper leans on

20 extracted references · 18 canonical work pages · cited by 1 Pith paper

[1]

Liu, and Dor Minzer

Amey Bhangale, Subhash Khot, Yang P. Liu, and Dor Minzer. On approximability of satisfiable k-csps: VII . 2024+

work page 2024
[2]

Liu, and Dor Minzer

Amey Bhangale, Subhash Khot, Yang P. Liu, and Dor Minzer. Reasonable bounds for combinatorial lines of length three. 2024+

work page 2024
[3]

On approximability of satisfiable k-csps: I

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: I . In STOC 2022 , pages 976--988. ACM , 2022

work page 2022
[4]

Effective bounds for restricted 3 -arithmetic progressions in F _p^n

Amey Bhangale, Subhash Khot, and Dor Minzer. Effective bounds for restricted 3 -arithmetic progressions in F _p^n . Electron. Colloquium Comput. Complex. , TR23-116 , 2023. To appear in Discrete Analysis

work page 2023
[5]

On approximability of satisfiable k -csps: II

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k -csps: II . In STOC 2023 , pages 632--642. ACM , 2023

work page 2023
[6]

On approximability of satisfiable k-csps: III

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: III . In STOC 2023 , pages 643--655. ACM , 2023

work page 2023
[7]

On approximability of satisfiable k-csps: IV

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: IV . In STOC 2024 , pages 1423--1434. ACM , 2024

work page 2024
[8]

On Approximability of Satisfiable k-CSPs: V

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: V . CoRR , abs/2408.15377, 2024

work page Pith review arXiv 2024

Show all 20 references

[9]

Direct sum testing

Roee David, Irit Dinur, Elazar Goldenberg, Guy Kindler, and Igor Shinkar. Direct sum testing. In ITCS 2015 , pages 327--336, 2015

2015
[10]

Direct sum testing: The general case

Irit Dinur and Konstantin Golubev. Direct sum testing: The general case. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques , 2019

2019
[11]

A density version of the H ales- J ewett theorem

Hillel Furstenberg and Yitzhak Katznelson. A density version of the H ales- J ewett theorem. Journal d’Analyse Math \'e matique , 57(1):64--119, 1991

1991
[12]

W. T. Gowers. A new proof of S zemer\'edi's theorem. Geom. Funct. Anal. , 11(3):465--588, 2001

2001
[13]

100 open problems

Ben Green. 100 open problems. manuscript
[14]

Regularity and positional games

AW Hales and RI Jewett. Regularity and positional games. Transactions of the American Mathematical Society , 106(2):222--229, 1963

1963
[15]

On subsets of finite abelian groups with no 3-term arithmetic progressions

Roy Meshulam. On subsets of finite abelian groups with no 3-term arithmetic progressions. Journal of Combinatorial Theory, Series A , 71(1):168--172, 1995

1995
[16]

Gaussian bounds for noise correlation of functions

Elchanan Mossel. Gaussian bounds for noise correlation of functions. Geometric and Functional Analysis , 19(6):1713--1756, 2010

2010
[17]

Analysis of boolean functions

Ryan O'Donnell. Analysis of boolean functions . Cambridge University Press, 2014

2014
[18]

A new proof of the density H ales- J ewett theorem

DHJ Polymath. A new proof of the density H ales- J ewett theorem. Annals of Mathematics , pages 1283--1327, 2012

2012
[19]

On certain sets of integers

Klaus F Roth. On certain sets of integers. J. London Math. Soc , 28(104-109):3, 1953

1953
[20]

New direct sum tests

Alek Westover, Edward Yu, and Kai Zheng. New direct sum tests. arXiv preprint arXiv:2409.10464 , 2024. to appear in ITCS 2025

2024 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Liu, and Dor Minzer

Amey Bhangale, Subhash Khot, Yang P. Liu, and Dor Minzer. On approximability of satisfiable k-csps: VII . 2024+

work page 2024

[2] [2]

Liu, and Dor Minzer

Amey Bhangale, Subhash Khot, Yang P. Liu, and Dor Minzer. Reasonable bounds for combinatorial lines of length three. 2024+

work page 2024

[3] [3]

On approximability of satisfiable k-csps: I

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: I . In STOC 2022 , pages 976--988. ACM , 2022

work page 2022

[4] [4]

Effective bounds for restricted 3 -arithmetic progressions in F _p^n

Amey Bhangale, Subhash Khot, and Dor Minzer. Effective bounds for restricted 3 -arithmetic progressions in F _p^n . Electron. Colloquium Comput. Complex. , TR23-116 , 2023. To appear in Discrete Analysis

work page 2023

[5] [5]

On approximability of satisfiable k -csps: II

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k -csps: II . In STOC 2023 , pages 632--642. ACM , 2023

work page 2023

[6] [6]

On approximability of satisfiable k-csps: III

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: III . In STOC 2023 , pages 643--655. ACM , 2023

work page 2023

[7] [7]

On approximability of satisfiable k-csps: IV

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: IV . In STOC 2024 , pages 1423--1434. ACM , 2024

work page 2024

[8] [8]

On Approximability of Satisfiable k-CSPs: V

Amey Bhangale, Subhash Khot, and Dor Minzer. On approximability of satisfiable k-csps: V . CoRR , abs/2408.15377, 2024

work page Pith review arXiv 2024