arXiv subjects

Subhash Khot

Publications and source records attributed to Subhash Khot.

At least 19 recordsLinked to original sources

arXiv2026

A Counting Lemma for Somewhat Restricted 3-APs

For a prime $p\geq 3$, a somewhat restricted $3$-AP in $\mathbb{F}_p^n$ is a triplet $(x,x+a,x+2a)$, where $x\in\mathbb{F}_p^n$ and $a\in \{0,1,2\}^n$. We prove a counting lemma for somewhat restricted $3$-APs in dense sets in $\mathbb{F}_p^n$. More precisely, we prove that for all $\alpha>0$, there exists $\beta>0$, such that for sufficiently large $n$, if a set $A\subseteq \mathbb{F}_p^n$ has density at least $\alpha$, then it contains at least $\beta$ fraction of all somewhat restricted $3$-APs. Our proof builds on recently developed machinery from [Bhangale, Khot, Minzer, 2026]. Our main new ingredient is an arithmetic regularity lemma for patterns such as somewhat restricted 3-APs. This result is in the spirit of arithmetic regularity lemmas from the theory of Gowers uniformity norms [Green, Tao, 2010] and may be of independent interest.

math.CO↗

arXiv2026

Conditioned free-energy density of proteins using unbalanced solutions to constraint satisfaction problems

We show that computing the log-partition function (free-energy) of conditioned inhomogeneous Curie--Weiss spin Hamiltonians reduces to an unbalanced $2 \to 1$ norm computation, and design a polynomial-time SDP algorithm for this problem with a lower bound proof for the amount of unbalance achieved. Applied to the protein Ubiquitin, the framework starts from a known crystal structure, explores alternative backbone conformations across the free-energy landscape, and identifies flexible regions of the protein while preserving its native secondary structure.

cs.LG↗

arXiv2025

An Analytical Approach to Parallel Repetition via CSP Inverse Theorems

Let $\mathcal{G}$ be a $k$-player game with value $<1$, whose query distribution is such that no marginal on $k-1$ players admits a non-trivial Abelian embedding. We show that for every $n\geq N$, the value of the $n$-fold parallel repetition of $\mathcal{G}$ is $$ \text{val}(\mathcal{G}^{\otimes n}) \leq \frac{1}{\underbrace{\log\log\cdots\log}_{C\text{ times}} n}, $$ where $N=N(\mathcal{G})$ and $1\leq C\leq k^{O(k)}$ are constants. As a consequence, we obtain a parallel repetition theorem for all $3$-player games whose query distribution is pairwise-connected. Prior to our work, only inverse Ackermann decay bounds were known for such games [Ver96]. As additional special cases, we obtain a unified proof for all known parallel repetition theorems, albeit with weaker bounds: (1) A new analytic proof of parallel repetition for all 2-player games [Raz98, Hol09, DS14]. (2) A new proof of parallel repetition for all $k$-player playerwise connected games [DHVY17, GHMRZ22]. (3) Parallel repetition for all $3$-player games (in particular $3$-XOR games) whose query distribution has no non-trivial Abelian embedding into $(\mathbb{Z}, +)$ [BKM23c, BBKLM25]. (4) Parallel repetition for all 3-player games with binary inputs [HR20, GHMRZ21, GHMRZ22, GMRZ22].

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arXiv2025

Biased Linearity Testing in the 1% Regime

We study linearity testing over the $p$-biased hypercube $(\{0,1\}^n, \mu_p^{\otimes n})$ in the 1% regime. For a distribution $\nu$ supported over $\{x\in \{0,1\}^k:\sum_{i=1}^k x_i=0 \text{ (mod 2)} \}$, with marginal distribution $\mu_p$ in each coordinate, the corresponding $k$-query linearity test $\text{Lin}(\nu)$ proceeds as follows: Given query access to a function $f:\{0,1\}^n\to \{-1,1\}$, sample $(x_1,\dots,x_k)\sim \nu^{\otimes n}$, query $f$ on $x_1,\dots,x_k$, and accept if and only if $\prod_{i\in [k]}f(x_i)=1$. Building on the work of Bhangale, Khot, and Minzer (STOC '23), we show, for $0 < p \leq \frac{1}{2}$, that if $k \geq 1 + \frac{1}{p}$, then there exists a distribution $\nu$ such that the test $\text{Lin}(\nu)$ works in the 1% regime; that is, any function $f:\{0,1\}^n\to \{-1,1\}$ passing the test $\text{Lin}(\nu)$ with probability $\geq \frac{1}{2}+\epsilon$, for some constant $\epsilon > 0$, satisfies $\Pr_{x\sim \mu_p^{\otimes n}}[f(x)=g(x)] \geq \frac{1}{2}+\delta$, for some linear function $g$, and a constant $\delta = \delta(\epsilon)>0$. Conversely, we show that if $k < 1+\frac{1}{p}$, then no such test $\text{Lin}(\nu)$ works in the 1% regime. Our key observation is that the linearity test $\text{Lin}(\nu)$ works if and only if the distribution $\nu$ satisfies a certain pairwise independence property.

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arXiv2024

On Approximability of Satisfiable $k$-CSPs: VI

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.

cs.CC↗

arXiv2024

On Approximability of Satisfiable $k$-CSPs: VII

Let $\Sigma_1,\ldots,\Sigma_k$ be finite alphabets, and let $\mu$ be a distribution over $\Sigma_1 \times \dots \times \Sigma_k$ in which the probability of each atom is at least $\alpha$. We prove that if $\mu$ does not admit Abelian embeddings, and $f_i: \Sigma_i \to \mathbb{C}$ are $1$-bounded functions (for $i=1,\ldots,k$) such that \[ \left|\mathbb{E}_{(x_1,\dots,x_k) \sim \mu^{\otimes n}}\Big[f_1(x_1) \dots f_k(x_k)\Big]\right| \geq \varepsilon, \] then there exists $L\colon \Sigma_1^n\to\mathbb{C}$ of degree at most $d$ and $\|L\|_2\leq 1$ such that $|\langle f_1, L\rangle|\geq \delta$, where $d$ and $\delta>0$ depend only on $k, \alpha$ and $\varepsilon$. This answers the analytic question posed by Bhangale, Khot, and Minzer (STOC 2022). We also prove several extensions of this result that are useful in subsequent applications.

cs.CC↗

arXiv2024

Reasonable Bounds for Combinatorial Lines of Length Three

We prove that any subset $A \subseteq [3]^n$ with $3^{-n}|A| \ge (\log\log\log\log n)^{-c}$ contains a combinatorial line of length $3$, i.e., $x, y, z \in A$, not all equal, with $x_i=y_i=z_i$ or $(x_i,y_i,z_i)=(0,1,2)$ for all $i = 1, 2, \dots, n$. This improves on the previous best bound of $3^{-n}|A| \ge \Omega((\log^* n)^{-1/2})$ of [D.H.J. Polymath, Ann. of Math. 2012].

math.CO↗

arXiv2024

On Approximability of Satisfiable k-CSPs: V

We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs. Our framework is based on a new hybrid approximation algorithm, which uses a combination of the Gaussian elimination technique (i.e., solving a system of linear equations over an Abelian group) and the semidefinite programming relaxation. We complement our algorithm with a matching dictator vs. quasirandom test that has perfect completeness. The analysis of our dictator vs. quasirandom test is based on a novel invariance principle, which we call the mixed invariance principle. Our mixed invariance principle is an extension of the invariance principle of Mossel, O'Donnell and Oleszkiewicz [Annals of Mathematics, 2010] which plays a crucial role in Raghavendra's work. The mixed invariance principle allows one to relate 3-wise correlations over discrete probability spaces with expectations over spaces that are a mixture of Guassian spaces and Abelian groups, and may be of independent interest.

cs.CC↗

arXiv2024

Parallel Repetition for $3$-Player XOR Games

In a $3$-$\mathsf{XOR}$ game $\mathcal{G}$, the verifier samples a challenge $(x,y,z)\sim \mu$ where $\mu$ is a probability distribution over $\Sigma\times\Gamma\times\Phi$, and a map $t\colon \Sigma\times\Gamma\times\Phi\to\mathcal{A}$ for a finite Abelian group $\mathcal{A}$ defining a constraint. The verifier sends the questions $x$, $y$ and $z$ to the players Alice, Bob and Charlie respectively, receives answers $f(x)$, $g(y)$ and $h(z)$ that are elements in $\mathcal{A}$ and accepts if $f(x)+g(y)+h(z) = t(x,y,z)$. The value, $\mathsf{val}(\mathcal{G})$, of the game is defined to be the maximum probability the verifier accepts over all players' strategies. We show that if $\mathcal{G}$ is a $3$-$\mathsf{XOR}$ game with value strictly less than $1$, whose underlying distribution over questions $\mu$ does not admit Abelian embeddings into $(\mathbb{Z},+)$, then the value of the $n$-fold repetition of $\mathcal{G}$ is exponentially decaying. That is, there exists $c=c(\mathcal{G})>0$ such that $\mathsf{val}(\mathcal{G}^{\otimes n})\leq 2^{-cn}$. This extends a previous result of [Braverman-Khot-Minzer, FOCS 2023] showing exponential decay for the GHZ game. Our proof combines tools from additive combinatorics and tools from discrete Fourier analysis.

cs.CC↗

arXiv2023

Parallel Repetition of k-Player Projection Games

We study parallel repetition of k-player games where the constraints satisfy the projection property. We prove exponential decay in the value of a parallel repetition of projection games with value less than 1.

cs.CC↗

arXiv2023

Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$

For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq \mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such a set is at most $\frac{C}{(\log\log\log n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/\log^{*} n)$, which follows from the density Hales-Jewett theorem.

math.CO↗

arXiv2023

On Approximability of Satisfiable k-CSPs: IV

We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $\Sigma,\Gamma$ and $\Phi$ are alphabets of constant size, and $\mu$ is a pairwise connected distribution over $\Sigma\times\Gamma\times\Phi$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $\Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon \Sigma^n\to\mathbb{C}$, $g\colon \Gamma^n\to\mathbb{C}$, $h\colon \Phi^n\to\mathbb{C}$ satisfying \[ \left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon \] must arise from an Abelian group associated with the distribution $\mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $\tilde{f}(x) = \chi(\sigma(x_1),\ldots,\sigma(x_n)) L (x)$, where $\sigma\colon \Sigma \to H$ is some map, $\chi\in \hat{H}^{\otimes n}$ is a character, and $L\colon \Sigma^n\to\mathbb{C}$ is a low-degree function with bounded $2$-norm. En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.

cs.CC↗

arXiv2022

Parallel Repetition for the GHZ Game: Exponential Decay

We show that the value of the $n$-fold repeated GHZ game is at most $2^{-\Omega(n)}$, improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup type questions from additive combinatorics.

cs.CC↗

arXiv2022

Improved Monotonicity Testers via Hypercube Embeddings

We show improved monotonicity testers for the Boolean hypercube under the $p$-biased measure, as well as over the hypergrid $[m]^n$. Our results are: 1. For any $p\in (0,1)$, for the $p$-biased hypercube we show a non-adaptive tester that makes $\tilde{O}(\sqrt{n}/\varepsilon^2)$ queries, accepts monotone functions with probability $1$ and rejects functions that are $\varepsilon$-far from monotone with probability at least $2/3$. 2. For all $m\in\mathbb{N}$, we show an $\tilde{O}(\sqrt{n}m^3/\varepsilon^2)$ query monotonicity tester over $[m]^n$. We also establish corresponding directed isoperimetric inequalities in these domains. Previously, the best known tester due to Black, Chakrabarty and Seshadhri had $\Omega(n^{5/6})$ query complexity. Our results are optimal up to poly-logarithmic factors and the dependency on $m$. Our proof uses a notion of monotone embeddings of measures into the Boolean hypercube that can be used to reduce the problem of monotonicity testing over an arbitrary product domains to the Boolean cube. The embedding maps a function over a product domain of dimension $n$ into a function over a Boolean cube of a larger dimension $n'$, while preserving its distance from being monotone; an embedding is considered efficient if $n'$ is not much larger than $n$, and we show how to construct efficient embeddings in the above mentioned settings.

cs.CC↗

arXiv2021

Almost Polynomial Factor Inapproximability for Parameterized k-Clique

The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for the k-Clique problem. Although, the non-existence of an F(k)-FPT-approximation algorithm for any computable sublinear function F is known under gap-ETH [Chalermsook et al., FOCS 2017], it has remained a long standing open problem to prove the same inapproximability result under the more standard and weaker assumption, W[1]$\neq$FPT. In a recent breakthrough, Lin [STOC 2021] ruled out constant factor (i.e., F(k)=O(1)) FPT-approximation algorithms under W[1]$\neq$FPT. In this paper, we improve this inapproximability result (under the same assumption) to rule out every $F(k)=k^{1/H(k)}$ factor FPT-approximation algorithm for any increasing computable function H (for example $H(k)=\log^\ast k$). Our main technical contribution is introducing list decoding of Hadamard codes over large prime fields into the proof framework of Lin.

cs.CC↗

arXiv2021

An Invariance Principle for the Multi-slice, with Applications

Given an alphabet size $m\in\mathbb{N}$ thought of as a constant, and $\vec{k} = (k_1,\ldots,k_m)$ whose entries sum of up $n$, the $\vec{k}$-multi-slice is the set of vectors $x\in [m]^n$ in which each symbol $i\in [m]$ appears precisely $k_i$ times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space $([m]^n,\mu^n)$ in which $\mu(i) = k_i/n$. This answers a question raised by Filmus et al. As applications of the invariance principle, we show: 1. An analogue of the "dictatorship test implies computational hardness" paradigm for problems with perfect completeness, for a certain class of dictatorship tests. Our computational hardness is proved assuming a recent strengthening of the Unique-Games Conjecture, called the Rich $2$-to-$1$ Games Conjecture. Using this analogue, we show that assuming the Rich $2$-to-$1$ Games Conjecture, (a) there is an $r$-ary CSP $\mathcal{P}_r$ for which it is NP-hard to distinguish satisfiable instances of the CSP and instances that are at most $\frac{2r+1}{2^r} + o(1)$ satisfiable, and (b) hardness of distinguishing $3$-colorable graphs, and graphs that do not contain an independent set of size $o(1)$. 2. A reduction of the problem of studying expectations of products of functions on the multi-slice to studying expectations of products of functions on correlated, product spaces. In particular, we are able to deduce analogues of the Gaussian bounds from \cite{MosselGaussian} for the multi-slice. 3. In a companion paper, we show further applications of our invariance principle in extremal combinatorics, and more specifically to proving removal lemmas of a wide family of hypergraphs $H$ called $\zeta$-forests, which is a natural extension of the well-studied case of matchings.

cs.CC↗

arXiv2020

Optimal Inapproximability of Satisfiable $k$-LIN over Non-Abelian Groups

A seminal result of H\r{a}stad [J. ACM, 48(4):798--859, 2001] shows that it is NP-hard to find an assignment that satisfies $\frac{1}{|G|}+\varepsilon$ fraction of the constraints of a given $k$-LIN instance over an abelian group, even if there is an assignment that satisfies $(1-\varepsilon)$ fraction of the constraints, for any constant $\varepsilon>0$. Engebretsen et al. [Theoretical Computer Science, 312(1):17--45, 2004] later showed that the same hardness result holds for $k$-LIN instances over any finite non-abelian group. Unlike the abelian case, where we can efficiently find a solution if the instance is satisfiable, in the non-abelian case, it is NP-complete to decide if a given system of linear equations is satisfiable or not, as shown by Goldmann and Russell [Information and Computation, 178(1):253--262. 2002]. Surprisingly, for certain non-abelian groups $G$, given a satisfiable $k$-LIN instance over $G$, one can in fact do better than just outputting a random assignment using a simple but clever algorithm. The approximation factor achieved by this algorithm varies with the underlying group. In this paper, we show that this algorithm is {\em optimal} by proving a tight hardness of approximation of satisfiable $k$-LIN instance over {\em any} non-abelian $G$, assuming $P \neq NP$. As a corollary, we also get $3$-query probabilistically checkable proofs with perfect completeness over large alphabets with improved soundness.

cs.CC↗

arXiv2018

The Andoni--Krauthgamer--Razenshteyn characterization of sketchable norms fails for sketchable metrics

Andoni, Krauthgamer and Razenshteyn (AKR) proved (STOC 2015) that a finite-dimensional normed space $(X,\|\cdot\|_X)$ admits a $O(1)$ sketching algorithm (namely, with $O(1)$ sketch size and $O(1)$ approximation) if and only if for every $\varepsilon\in (0,1)$ there exist $\alpha\geqslant 1$ and an embedding $f:X\to \ell_{1-\varepsilon}$ such that $\|x-y\|_X\leqslant \|f(x)-f(y)\|_{1-\varepsilon}\leqslant \alpha \|x-y\|_X$ for all $x,y\in X$. The "if part" of this theorem follows from a sketching algorithm of Indyk (FOCS 2000). The contribution of AKR is therefore to demonstrate that the mere availability of a sketching algorithm implies the existence of the aforementioned geometric realization. Indyk's algorithm shows that the "if part" of the AKR characterization holds true for any metric space whatsoever, i.e., the existence of an embedding as above implies sketchability even when $X$ is not a normed space. Due to this, a natural question that AKR posed was whether the assumption that the underlying space is a normed space is needed for their characterization of sketchability. We resolve this question by proving that for arbitrarily large $n\in \mathbb{N}$ there is an $n$-point metric space $(M(n),d_{M(n)})$ which is $O(1)$-sketchable yet for every $\varepsilon\in (0,\frac12)$, if $\alpha(n)\geqslant 1$ and $f_n:M(n)\to \ell_{1-\varepsilon}$ are such that $d_{M(n)}(x,y)\leqslant \|f_n(x)-f_n(y)\|_{1-\varepsilon}\leqslant \alpha(n) d_{M(n)}(x,y)$ for all $x,y\in M(n)$, then necessarily $\lim_{n\to \infty} \alpha(n)= \infty$.

cs.DS↗