arXiv subjects

Zejun Huang

Publications and source records attributed to Zejun Huang.

At least 19 recordsLinked to original sources

arXiv2026

Triton for MTIA: Bridging the Programming Model Gaps for Custom AI Accelerators

The rapid growth in machine learning workloads has fueled the proliferation of custom accelerator architectures. Designed from the ground up, these accelerators often expose programming models that are distinct from GPUs. While hyperscalers and AI chip startups continue to innovate in this space, achieving broad operator coverage to support diverse models remains a major challenge. Additionally, an easy-to-use, high-level kernel programming language is important for rapid iteration of models and kernels. Triton, together with TorchInductor, addresses these issues on GPUs, but its viability on accelerators with different programming models has yet to be established. In this work, we present the first production-scale application of Triton on a custom ML accelerator, MTIA-2i, developed by Meta. To support MTIA-2i, we develop a new compiler backend that targets it, introduce enhancements to TorchInductor code generation, and propose minimal language extensions that expose MTIA-specific architectural features. We demonstrate that Triton-MTIA kernels achieve performance competitive with expert-tuned C++ implementations. Leveraging these development efficiency gains, we successfully deployed manually written and Inductor-generated Triton kernels in production across approximately 60 different model types, accounting for 50% of layers and 47% of non-GEMM execution time for these models. Our results provide compelling evidence that DSLs like Triton can bridge the programming model gaps between ML frameworks, kernels, and custom accelerators, enabling rapid innovation and efficient deployment at scale.

cs.PL↗

arXiv2026

Three Results on Generalized Quasikernels in Digraphs

A $q$-kernel of a digraph $D$ is an independent set $Q\subseteq D$ such that every vertex of $D$ is reachable from $Q$ by a directed path of length at most $q$, which is a natural generalization of kernels and quasikernels. In this paper, we establish three results on generalized quasikernels. Firstly, we prove that any $n$-vertex source-free bipartite oriented graph with no directed 4-cycles has a quasikernel of size at most $17n/35$. Secondly, we show that every digraph with no $(r-1)$-source set contains $r$ pairwise disjoint $(3r-2)$-kernels, where $r\ge 2$. At last, we consider unicyclic digraph with a directed cycle of length $2\ell$ and bipartition $U\cup V$, and we prove that for every odd integer $q\ge 3$, there exist two $q$-kernels $Q_U\subseteq U$ and $Q_V\subseteq V$ such that \[ |Q_U|+|Q_V| \le 2\cdot \frac{\lceil \ell/(q+1)\rceil}{\ell} |V(D)|. \] These results confirm two conjectures and give an affirmative answer to a question posed by Spiro in European Journal of Combinatorics 133 (2026), 104307.

math.CO↗

arXiv2026

On a conjecture regarding the product version of the Hilton-Milner theorem

Recently, Frankl and Wang considered a product version of the classical Hilton-Milner theorem. They conjectured that, if $\mathcal{F} \subset \binom{[n]}{k}$ and $\mathcal{G} \subset \binom{[n]}{\ell}$ are non-trivial cross-intersecting families with $n \geq 2k > 2\ell \geq 4$, the maximum of $|\mathcal{F}||\mathcal{G}|$ is attained by the natural Hilton-Milner-type configurations. In this paper, we present two main results concerning this conjecture. Firstly, we show that the conjecture does not hold in general. By introducing a two-center construction, we prove that for every fixed integer $\ell \geq 3$ and all sufficiently large $k$, the conjecture is false in a linear range $2k+1 \leq n \leq (c_\ell - \epsilon)k$ for any $0 < \epsilon < c_\ell - 2$, where $c_\ell > 2$ is an explicit constant. Secondly, we prove that the conjecture holds when $n > 100\ell k^2$ and $3 \leq \ell < k$, and we completely characterize the extremal families. Our proofs rely on the size of minimal covers and analyzing the structural properties of $2$-cover graphs.

math.CO↗

arXiv2026

Connected graphs with a large dissociation number attaining the minimum spectral radius

A dissociation set in a graph is a subset of vertices that induces a subgraph of maximum degree at most one, which is a natural generalization of the notion of an independent set. The dissociation number of a graph is defined as the maximum cardinality of a dissociation set. This paper studies the minimum spectral radius of connected graphs with a given order $n$ and a given dissociation number $\psi$. For $\psi=n-k$ with $k\ge 4$ fixed and $n$ sufficiently large, we establish both upper and lower bounds for this minimum spectral radius and prove the extremal graphs must belong to a specific graph class.

math.CO↗

arXiv2026

Thresholds for the Frankl-Wang $3/7$ conjecture on maximum-degree ratios

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family, $\Delta(\mathcal{F})=\max_{x\in[n]}|\{F\in\mathcal{F}:x\in F\}|$, and $\varrho(\mathcal{F})=\Delta(\mathcal{F})/|\mathcal{F}|$. Frankl and Wang conjectured that if $n>100k$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$; the constant $3/7$ is sharp because of the Fano-plane construction. In this note we obtain three results. First, we show that no linear threshold $n>Ck$ can be sufficient: using a truncated Fano-plane construction we exhibit, for every constant $C$ and all large $k$, an intersecting family with $n>Ck$, $|\mathcal{F}|>\binom{n-3}{k-3}$, yet $\varrho(\mathcal{F})<3/7$. In particular, the original condition $n>100k$ does not guarantee the conclusion. Second, for $k=3$ we prove that $\varrho(\mathcal{F})\ge 3/7$ holds for every nonempty intersecting $3$-uniform family; the proof is nontrivial and does not rely on any assumption on $n$ or $|\mathcal{F}|$. Third, using the classical pseudo-sunflower bound $|\mathcal{F}|\le t^k$ (for families containing no pseudo-sunflower of size $t+1$), we obtain a completely explicit polynomial threshold for all $k\ge4$: if $n>(k-3)(7k^4+k)+3$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$. In particular, the simplified bound $n>7k^5$ is sufficient for every $k\ge4$.

math.CO↗

arXiv2026

Improved bound on symmetric differences of intersecting families

For a family $\mathcal{F}$, it is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in\mathcal{F}$. We use $\mathcal{SD}(\mathcal{F}) = \{F \triangle G : F, G \in \mathcal{F}\}$ to denote the family of symmetric differences of $\mathcal{F}$. In 2023, Frankl, Kiselev and Kupavskii conjectured that for any intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n > 10k$, the inequality $|\mathcal{SD}(\mathcal{F})| \le \sum_{\ell=0}^{k-1} \binom{n-1}{2\ell}$ holds. They further observed that a proof for the range $n>3k^2$ could likely be obtained via arguments similar to those in their earlier work, though no detailed derivation was given. In this paper, we establish the conjecture under the conditions $n\ge 100k\ln k$ and $k\ge 50$. We also determine the extremal families, which are precisely a certain class of stars. A concentration inequality plays a central role in the proof.

math.CO↗

arXiv2026

Connected graphs minimizing the spectral radius for given order and dissociation number

A dissociation set in a graph is a subset of vertices which induces a subgraph with maximum degree at most one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we consider the $n$-vertex connected graphs with a given dissociation number that attain the minimum spectral radius. By using structure analysis and constructing difference equations, we characterize the extremal graphs with dissociation number $n-3$.

math.CO↗

arXiv2025

The dimension and Bose distance of some BCH codes of length $\frac{q^{m}-1}{\lambda}$

BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length $(q^m - 1)/\lambda$ over the finite field $\mathbb{F}_q$, where $\lambda$ is a positive divisor of $q - 1$. Specifically, for narrow-sense BCH codes of this length with $m \geq 4$, we derive explicit formulas for their dimension for designed distance $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda} + 1$. We also provide explicit formulas for their Bose distance in the range $2 \leq \delta \leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/{\lambda}$. These ranges for $\delta$ are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Several optimal linear codes can be obtained from these BCH codes.

cs.IT↗

arXiv2025

Soliton Surfaces and the Geometry of Integrable Deformations of the $\mathbb{CP}^{N-1}$ Model

The $\mathbb{CP}^{N-1}$ model is an analytically tractable $2d$ quantum field theory which shares several properties with $4d$ Yang-Mills theory. By virtue of its classical integrability, this model also admits a family of integrable higher-spin auxiliary field deformations, including the $T \overline{T}$ deformation as a special case. We study the $\mathbb{CP}^{N-1}$ model and its deformations from a geometrical perspective, constructing their soliton surfaces and recasting physical properties of these theories as statements about surface geometry. We examine how the $T \overline{T}$ flow affects the unit constraint in the $\mathbb{CP}^{N-1}$ model and prove that any solution of this theory with vanishing energy-momentum tensor remains a solution under analytic stress tensor deformations -- an argument that extends to generic dimensions and instanton-like solutions in stress tensor flows including the non-analytic, $2d$, root-$T \overline{T}$ case and classes of higher-spin, Smirnov-Zamolodchikov-type, deformations. Finally, we give two geometric interpretations for general $T \overline{T}$-like deformations of symmetric space sigma models, showing that such flows can be viewed as coupling the undeformed theory to a unit-determinant field-dependent metric, or using a particular choice of moving frame on the soliton surface.

hep-th↗

arXiv2025

The dimension and Bose distance of certain primitive BCH codes

BCH codes are a significant class of cyclic codes that play an important role in both theoretical research and practical applications. Their strong error-correcting abilities and efficient encoding and decoding methods make BCH codes widely applicable in various areas, including communication systems, data storage devices, and consumer electronics. Although BCH codes have been extensively studied, the parameters of BCH codes are not known in general. Let $q$ be a prime power and $m$ be a positive integer. Denote by $\mathcal{C}_{\left(q,m,\delta)\right)}$ the narrow-sense primitive BCH code with length $q^m-1$ and designed distance $\delta$. As of now, the dimensions of $\mathcal{C}_{(q,m,\delta)}$ are fully understood only for $m \leq 2$. For $m \geq 4$, the dimensions of $\mathcal{C}_{(q,m,\delta)}$ are known only for the range $2 \leq \delta \leq q^{\lfloor (m+1)/2 \rfloor +1}$ and for a limited number of special cases. In this paper, we determined the dimension and Bose distance of $\mathcal{C}_{(q,m,\delta)}$ for $m\geq 4$ and $\delta\in [2, q^{\lfloor ( 2m-1)/{3}\rfloor+1}]. $ Additionally, we have also extended our results to some primitive BCH codes that are not necessarily narrow-sense.

cs.IT↗

arXiv2024

Nonregular graphs with a given maximum degree attaining maximum spectral radius

Let $G$ be a connected nonregular graphs of order $n$ with maximum degree $\Delta$ that attains the maximum spectral radius. Liu and Li (2008) proposed a conjecture stating that $G$ has a degree sequence $(\Delta,\ldots,\Delta,\delta)$ with $\delta<\Delta$. For $\Delta=3$ and $\Delta=4$, Liu (2024) confirmed this conjecture by characterizing the structure of such graphs. Liu also proposed a modified version of the conjecture for fixed $\Delta$ and sufficiently large $n$, stating that the above $\delta=\Delta-1$ if $\Delta$ and $n$ are both odd, $\delta=1$ if $\Delta$ is odd and $n$ is even, and $\delta=\Delta-2$ if $\Delta$ is even. For the cases where $\Delta=n-2$ with $n\ge 5$, and $\Delta=n-3$ with $n\ge 59$, we fully characterize the structure of $G$.

math.CO↗

arXiv2024

Extremal digraphs containing at most $t$ paths of length 2 with the same endpoints

Given a positive integer $t$, let $P_{t,2}$ be the digraph consisting of $t$ directed paths of length 2 with the same initial and terminal vertices. In this paper, we study the maximum size of $P_{t+1,2}$-free digraphs of order $n$, which is denoted by $ex(n, P_{t+1,2})$. For sufficiently large $n$, we prove that $ex(n, P_{t+1})=g(n,t)$ when $\lfloor(n-t)/{2} \rfloor$ is odd and $ex(n, P_{t+1,2})\in \{g(n,t)-1, g(n,t)\}$ when $\lfloor(n-t)/{2} \rfloor$ is even, where $g(n,t)=\left\lceil(n+t)/{2}\right\rceil \left\lfloor(n-t)/{2}\right\rfloor+tn+1$.

math.CO↗

arXiv2024

Extremal oriented graphs avoiding 1-subdivision of an in-star

An oriented graph is a digraph obtained from an undirected graph by choosing an orientation for each edge. Given a positive integer $n$ and an oriented graph $F$, the oriented Tur$\acute{\rm a}$n number $ex_{ori}(n,F)$ is the maximum number of arcs in an $F$-free oriented graph of order $n$. In this paper, we investigate the oriented Tur$\acute{\rm a}$n number $ex_{ori}(n, \overrightarrow{S_{k,1}} )$, where $\overrightarrow{S_{k,1}}$ is the $1$-subdivision of the in-star of order $k+1$. We determine $ex_{ori}(n,\overrightarrow{S_{k,1}}) $ for $k=2,3$ as well as the extremal oriented graphs. For $k\ge 4$, we establish a lower bound and an upper bound on $ex_{ori}(n,\overrightarrow{S_{k,1}})$.

math.CO↗

arXiv2024

The maximum number of maximum dissociation sets in potted graphs

A potted graph is a unicyclic graph such that its cycle contains a unique vertex with degree larger than 2. Given a graph $G$, a subset of $V(G)$ is a dissociation set of $G$ if it induces a subgraph with maximum degree at most one. A maximum dissociation set is a dissociation set with maximum cardinality. In this paper, we determine the maximum number of maximum dissociation sets in a potted graph of order $n$ which contains a fixed cycle. Extremal potted graphs attaining this maximum number are also characterized.

math.CO↗

arXiv2023

Connected graphs with a given dissociation number attaining the minimum spectral radius

A dissociation set of a graph is a set of vertices which induces a subgraph with maximum degree less than or equal to one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we study the connected graphs of order $n$ with a given dissociation number that attains the minimum spectral radius. We characterize these graphs when the dissociation number is in $\{n-1,~n-2,~\lceil2n/3\rceil,~\lfloor2n/3\rfloor,~2\}$. We also prove that these graphs are trees when the dissociation number is larger than $\lceil {2n}/{3}\rceil$.

math.CO↗

arXiv2023

Linear maps preserving (p,k) norms of tensor products of matrices

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices. Let $\|\cdot\|_{(p,k)}$ be the $(p,k)$ norm on $M_{mn}$ with $1\leq k\leq mn$ and $2<p<\infty$. We show that a linear map $\phi:M_{mn}\rightarrow M_{mn}$ satisfies $$\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} {\rm\quad for~ all\quad}A\in M_m {\rm ~and ~}B\in M_n$$ if and only if there exist unitary matrices $U,V\in M_{mn}$ such that $$\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V {\rm\quad for~ all\quad}A\in M_m {\rm~ and~ }B\in M_n,$$ where $\varphi_s$ is the identity map or the transposition map $X\to X^T$ for $s=1,2$. The result is also extended to multipartite systems.

math.FA↗

arXiv2023

$T \overline{T}$-Like Flows and $3d$ Nonlinear Supersymmetry

We show that the $3d$ Born-Infeld theory can be generated via an irrelevant deformation of the free Maxwell theory. The deforming operator is constructed from the energy-momentum tensor and includes a novel non-analytic contribution that resembles root-$T \overline{T}$. We find that a similar operator deforms a free scalar into the scalar sector of the Dirac-Born-Infeld action, which describes transverse fluctuations of a D-brane, in any dimension. We also analyse trace flow equations and obtain flows for subtracted models driven by a relevant operator. In $3d$, the irrelevant deformation can be made manifestly supersymmetric by presenting the flow equation in $\mathcal{N} = 1$ superspace, where the deforming operator is built from supercurrents. We demonstrate that two supersymmetric presentations of the D2-brane effective action, the Maxwell-Goldstone multiplet and the tensor-Goldstone multiplet, satisfy superspace flow equations driven by this supercurrent combination. To do this, we derive expressions for the supercurrents in general classes of vector and tensor/scalar models by directly solving the superspace conservation equations and also by coupling to $\mathcal{N} = 1$ supergravity. As both of these multiplets exhibit a second, spontaneously broken supersymmetry, this analysis provides further evidence for a connection between current-squared deformations and nonlinearly realized symmetries.

hep-th↗

arXiv2022

Cliques and independent subgroups of the Birkhoff polytope graph

The Birkhoff polytope $\Omega_n$ is the polytope of doubly stochastic matrices of order $n$. The Birkhoff polytope graph $G(\Omega_n)$ is the skeleton of $\Omega_n$; it is the Cayley graph whose vertex set consists of the elements of the symmetric group ${\rm Sym}(n)$ of degree $n$, where two permutations are adjacent if one equals the product of the other with a cycle. We study the combinatorial structure of this graph, focusing on its maximal and maximum cliques and on its independent subgroups (subgroups of ${\rm Sym}(n)$ whose elements are pairwise nonadjacent in the graph). We obtain maximal subgroups of $G(\Omega_n)$ and establish both a lower bound and an upper bound for its clique number. Especially, we prove that if $K$ is a subset of ${\rm Sym}(n)$ consisting of 3-cycle permutations such that $\delta_1^{-1}\delta_2$ is a single cycle for all $\delta_1,\delta_2\in K$, then the maximum size of $K$ is $\lfloor (n-1)^2/4\rfloor$, which can be viewed as an Erd\H{o}s-Ko-Rado-type theorem for ${\rm Sym}(n)$.

math.CO↗