arXiv subjects
Publications and source records attributed to Yeyuan Chen.
We give a deterministic polynomial-time list-decoding algorithm for Reed-Solomon codes over prime fields that approaches list-decoding capacity for every evaluation set and every constant rate.
We extend the recent work of Reis and Rothvoss on sparsifying sums of $\ell_1$ norms to the more general task of sparsifying (Minkowski) sums of centrally symmetric, convex sets. As our main result, we prove that for any $\varepsilon > 0$ and centrally symmetric, convex sets $C_1, \ldots, C_m\subseteq\mathbb{R}^n$ there is a choice of weights $λ_1, \dots , λ_m \in \mathbb{R}_{\geq 0}$ such that at most $O(n / \varepsilon^2)$ of the weights are non-zero, and \[(1 - \varepsilon)\cdot C\subseteq\sum_{i = 1}^mλ_i\cdot C_i\subseteq(1 + \varepsilon)\cdot C,\] where $C:= C_1 + \cdots + C_m$ refers to the Minkowski sums of the sets $C_1, \ldots, C_m$, and $λ\cdot C$ refers to the dilation of the set $C$. As immediate applications of this result, we obtain sparsifiers of size $O(n / \varepsilon^2)$ for sparsifying sums of seminorms in $n$-dimensional space, improving on the $O\left ( \frac{n \log(n/\varepsilon) \cdot \log^{2.5}(n)}{\varepsilon^2} \right )$ size sparsifiers from the work of Jambulapati, Lee, Liu, and Sidford (FOCS 2023). This further yields optimal size hypergraph cut sparsifiers with $O(n / \varepsilon^2)$ hyperedges, improving on the $O(n \log(n) / \varepsilon^2)$ size sparsifiers from the work of Chen, Khanna, and Nagda (FOCS 2020). More generally, this also gives optimal size sparsifiers for sums of symmetric submodular functions.
In coding theory, list recoverability is a fundamental concept which robustly captures how ``spread-out'' codewords are in a code. More formally, given a code $C \subseteq Σ^n$ and input lists $S_1, \dots, S_n \subseteq Σ$ of size at most $\ell$, list recoverability requires that there are at most $L$ codewords $c \in C$ such that $c_i \in S_i$ for at least $(1-ρ)n$ choices of $i \in [n]$. List recovery is an important question which has found applications in many areas, including complexity theory, property testing, compressed sensing, streaming algorithms, and cryptography. As our first main result, we establish a tight ``generalized singleton bound''. Formally, we show that for constant $\ell, L,ρ$ and sufficiently large alphabets $Σ$, if we define $R^*=\frac{L+1-\ell}{L}-\frac{L+1}{L}ρ$, it is possible for a $(ρ,\ell,L)$ list-recoverable code to have rate $R^*-ε$ but impossible to have rate $R^*+ε$. One direction of our result already directly generalizes and improves a weaker impossibility result due to Goldberg, Shangguan, and Tamo. For our second main result, we prove that there is a fundamental shortcoming in existing methods that aim to construct explicit, optimal list-recoverable codes. Indeed, recent work has constructed explicit codes achieving list-decoding capacity (along with other related properties) using a framework introduced in the work of Alon--Edmonds--Luby (AEL). We give a meta-analysis of such constructions by presenting an ``AEL framework'' which captures all such recent constructions in the literature. Within this framework, we show that no AEL-based code can break a recently-identified list-recovery barrier for additive and linear codes.
Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed--Solomon (IRS), folded Reed--Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors. Our algorithm for interleaved Reed--Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher--Kiayias--Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed--Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami--Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.
In coding theory, the problem of list recovery asks one to find all codewords $c$ of a given code $C$ which such that at least $1-ρ$ fraction of the symbols of $c$ lie in some predetermined set of $\ell$ symbols for each coordinate of the code. A key question is bounding the maximum possible list size $L$ of such codewords for the given code $C$. In this paper, we give novel combinatorial bounds on the list recoverability of various families of linear and folded linear codes, including random linear codes, random Reed--Solomon codes, explicit folded Reed--Solomon codes, and explicit univariate multiplicity codes. Our main result is that in all of these settings, we show that for code of rate $R$, when $ρ= 1 - R - ε$ approaches capacity, the list size $L$ is at most $(\ell/(R+ε))^{O(R/ε)}$. These results also apply in the average-radius regime. Our result resolves a long-standing open question on whether $L$ can be bounded by a polynomial in $\ell$. In the zero-error regime, our bound on $L$ perfectly matches known lower bounds. The primary technique is a novel application of a discrete entropic Brascamp--Lieb inequality to the problem of list recovery, allowing us to relate the local structure of each coordinate with the global structure of the recovered list. As a result of independent interest, we show that a recent result by Chen and Zhang (STOC 2025) on the list decodability of folded Reed--Solomon codes can be generalized into a novel Brascamp--Lieb type inequality.
In coding theory, a common question is to understand the threshold rates of various local properties of codes, such as their list decodability and list recoverability. A recent work Levi, Mosheiff, and Shagrithaya (FOCS 2025) gave a novel unified framework for calculating the threshold rates of local properties for random linear and random Reed--Solomon codes. In this paper, we extend their framework to studying the local properties of subspace designable codes, including explicit folded Reed-Solomon and univariate multiplicity codes. Our first main result is a local equivalence between random linear codes and (nearly) optimal subspace design codes up to an arbitrarily small rate decrease. We show any local property of random linear codes applies to all subspace design codes. As such, we give the first explicit construction of folded linear codes that simultaneously attain all local properties of random linear codes. Conversely, we show that any local property which applies to all subspace design codes also applies to random linear codes. Our second main result is an application to matroid theory. We show that the correctable erasure patterns in a maximally recoverable tensor code can be identified in deterministic polynomial time, assuming a positive answer to a matroid-theoretic question due to Mason (1981). This improves on a result of Jackson and Tanigawa (JCTB 2024) who gave a complexity characterization of $\mathsf{RP} \cap \mathsf{coNP}$ assuming a stronger conjecture. Our result also applies to the generic bipartite rigidity and matrix completion matroids. As a result of additional interest, we study the existence and limitations of subspace designs. In particular, we tighten the analysis of family of subspace designs constructioned by Guruswami and Kopparty (Combinatorica 2016) and show that better subspace designs do not exist over algebraically closed fields.
In this paper, we prove that explicit FRS codes and multiplicity codes achieve relaxed generalized Singleton bounds for list size $L\ge1.$ Specifically, we show the following: (1) FRS code of length $n$ and rate $R$ over the alphabet $\mathbb{F}_q^s$ with distinct evaluation points is $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ list-decodable (LD) for list size $L\in[s]$. (2) Multiplicity code of length $n$ and rate $R$ over the alphabet $\mathbb{F}_p^s$ with distinct evaluation points is $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ LD for list size $L\in[s]$. Choosing $s=Θ(1/ε^2)$ and $L=O(1/ε)$, our results imply that both FRS codes and multiplicity codes achieve LD capacity $1-R-ε$ with optimal list size $O(1/ε)$. This exponentially improves the previous state of the art $(1/ε)^{O(1/ε)}$ established by Kopparty et. al. (FOCS 2018) and Tamo (IEEE TIT, 2024). In particular, our results on FRS codes fully resolve a open problem proposed by Guruswami and Rudra (STOC 2006). Furthermore, our results imply the first explicit constructions of $(1-R-ε,O(1/ε))$ LD codes of rate $R$ with poly-sized alphabets. Our method can also be extended to analyze the list-recoverability (LR) of FRS codes. We provide a tighter radius upper bound that FRS codes cannot be $(\frac{L+1-\ell}{L+1}(1-\frac{mR}{m-1})+o(1),\ell, L)$ LR where $m=\lceil\log_{\ell}{(L+1)}\rceil$. We conjecture this bound is almost tight when $L+1=\ell^a$ for any $a\in\mathbb{N}^{\ge 2}$. To give some evidences, we show FRS codes are $\left(\frac{1}{2}-\frac{sR}{s-2},2,3\right)$ LR, which proves the tightness in the smallest non-trivial case. Our bound refutes the possibility that FRS codes could achieve LR capacity $(1-R-ε, \ell, O(\frac{\ell}ε))$. This implies an intrinsic separation between LD and LR of FRS codes.
We construct constant-sized ensembles of linear error-correcting codes over any fixed alphabet that can correct a given fraction of adversarial erasures at rates approaching the Singleton bound arbitrarily closely. We provide several applications of our results: 1. Explicit constructions of strong linear seeded symbol-fixing extractors and lossless condensers, over any fixed alphabet, with only a constant seed length and optimal output lengths; 2. A strongly explicit construction of erasure codes on bipartite graphs (more generally, linear codes on matrices of arbitrary dimensions) with optimal rate and erasure-correction trade-offs; 3. A strongly explicit construction of erasure codes on non-bipartite graphs (more generally, linear codes on symmetric square matrices) achieving improved rates; 4. A strongly explicit construction of linear nearly-MDS codes over constant-sized alphabets that can be encoded and decoded in quasi-linear time.
The greedy algorithm adapted from Kruskal's algorithm is an efficient and folklore way to produce a $k$-spanner with girth at least $k+2$. The greedy algorithm has shown to be `existentially optimal', while it's not `universally optimal' for any constant $k$. Here, `universal optimality' means an algorithm can produce the smallest $k$-spanner $H$ given any $n$-vertex input graph $G$. However, how well the greedy algorithm works compared to `universal optimality' is still unclear for superconstant $k:=k(n)$. In this paper, we aim to give a new and fine-grained analysis of this problem in undirected unweighted graph setting. Specifically, we show some bounds on this problem including the following two (1) On the negative side, when $k<\frac{1}{3}n-O(1)$, the greedy algorithm is not `universally optimal'. (2) On the positive side, when $k>\frac{2}{3}n+O(1)$, the greedy algorithm is `universally optimal'. We also introduce an appropriate notion for `approximately universal optimality'. An algorithm is $(α,β)$-universally optimal iff given any $n$-vertex input graph $G$, it can produce a $k$-spanner $H$ of $G$ with size $|H|\leq n+α(|H^*|-n)+β$, where $H^*$ is the smallest $k$-spanner of $G$. We show the following positive bounds. (1) When $k>\frac{4}{7}n+O(1)$, the greedy algorithm is $(2,O(1))$-universally optimal. (2) When $k>\frac{12}{23}n+O(1)$, the greedy algorithm is $(18,O(1))$-universally optimal. (3) When $k>\frac{1}{2}n+O(1)$, the greedy algorithm is $(32,O(1))$-universally optimal. All our proofs are constructive building on new structural analysis on spanners. We give some ideas about how to break small cycles in a spanner to increase the girth. These ideas may help us to understand the relation between girth and spanners.
A $(d_1,d_2)$-biregular bipartite graph $G=(L\cup R,E)$ is called left-$(m,δ)$ unique-neighbor expander iff each subset $S$ of the left vertices with $|S|\leq m$ has at least $δd_1|S|$ unique-neighbors, where unique-neighbors mean vertices with exactly one neighbor in $S$. We can also define right/two-sided expanders similarly. In this paper, we give the following three strongly explicit constructions of unique-neighbor expanders with better unique-neighbor expansion for polynomial-sized sets, while sufficient expansion for linear-sized sets is also preserved: (1) Two-sided $(n^{1/3-ε},1-ε)$ lossless expanders for arbitrary $ε>0$ and aspect ratio. (2) Left-$(Ω(n),1-ε)$ lossless expanders with right-$(n^{1/3-ε},δ)$ expansion for some $δ>0$. (3) Two-sided-$(Ω(n),δ)$ unique-neighbor expanders with two-sided-$(n^{Ω(1)},1/2-ε)$ expansion. The second construction exhibits the first explicit family of one-sided lossless expanders with unique-neighbor expansion for polynomial-sized sets from the other side and constant aspect ratio. The third construction gives two-sided unique-neighbor expanders with additional $(1/2-ε)$ unique-neighbor expansion for two-sided polynomial-sized sets, which approaches the $1/2$ requirement in Lin and Hsieh (arXiv:2203.03581). Our techniques involve tripartite product recently introduced by Hsieh et al (STOC 2024), combined with a generalized existence argument of biregular graph with optimal two-sided unique-neighbor expansion for almost all degrees. We also use a new reduction from large girth/bicycle-freeness to vertex expansion, which might be of independent interest.
As a powerful framework for graph representation learning, Graph Neural Networks (GNNs) have garnered significant attention in recent years. However, to the best of our knowledge, there has been no formal analysis of the logical expressiveness of GNNs as Boolean node classifiers over multi-relational graphs, where each edge carries a specific relation type. In this paper, we investigate $\mathcal{FOC}_2$, a fragment of first-order logic with two variables and counting quantifiers. On the negative side, we demonstrate that the R$^2$-GNN architecture, which extends the local message passing GNN by incorporating global readout, fails to capture $\mathcal{FOC}_2$ classifiers in the general case. Nevertheless, on the positive side, we establish that R$^2$-GNNs models are equivalent to $\mathcal{FOC}_2$ classifiers under certain restricted yet reasonable scenarios. To address the limitations of R$^2$-GNNs regarding expressiveness, we propose a simple graph transformation technique, akin to a preprocessing step, which can be executed in linear time. This transformation enables R$^2$-GNNs to effectively capture any $\mathcal{FOC}_2$ classifiers when applied to the "transformed" input graph. Moreover, we extend our analysis of expressiveness and graph transformation to temporal graphs, exploring several temporal GNN architectures and providing an expressiveness hierarchy for them. To validate our findings, we implement R$^2$-GNNs and the graph transformation technique and conduct empirical tests in node classification tasks against various well-known GNN architectures that support multi-relational or temporal graphs. Our experimental results consistently demonstrate that R$^2$-GNN with the graph transformation outperforms the baseline methods on both synthetic and real-world datasets