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Hung Le

Publications and source records attributed to Hung Le.

At least 55 records · Page 3Linked to original sources

Fine-Grained Complexity of Continuous Euclidean k-Center

In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

cs.CG↗

Optimal Bounds for Spanners and Tree Covers in Doubling Metrics

It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+ε)$-spanner with maximum degree $\tilde{O}(ε^{-d+1})$ and with lightness $\tilde{O}(ε^{-d})$; 2) a $(1+ε)$-tree cover with $\tilde{O}(n \cdot ε^{-d+1})$ trees and maximum degree of $O(1)$ in each tree. Moreover, all the parameters in these constructions are optimal: there exists an $n$-point set in $\mathbb{R}^d$, for which any $(1+ε)$-spanner has $\tildeΩ(n \cdot ε^{-d+1})$ edges and lightness $\tildeΩ(ε^{-d})$. The upper bounds for Euclidean spanners rely heavily on the spatial property of cone partitioning in $\mathbb{R}^d$, which does not seem to extend to the wider family of doubling metrics, i.e., metric spaces of constant doubling dimension. In doubling metrics, a simple spanner construction from two decades ago, the net-tree spanner, has $\tilde{O}(n \cdot ε^{-d})$ edges, and it could be transformed into a spanner of maximum degree $\tilde{O}(ε^{-d})$ and lightness $\tilde{O}(n \cdot ε^{-(d+1)})$ by pruning redundant edges. Moreover, a careful refinement of the net-tree spanner yields a $(1+ε)$-tree cover with $\tilde{O}(ε^{-d})$ trees. Despite a large body of work, the problem of obtaining tight bounds for spanners and tree covers in the wider family of doubling metrics has remained elusive. We resolve this problem by presenting: 1) a surprisingly simple and tight lower bound, which shows that the net-tree spanner and its pruned version are optimal with respect to all the involved parameters, 2) a new construction of $(1+ε)$-tree covers with $\tilde{O}(n \cdot ε^{-d})$ trees, with maximum degree $O(1)$ in each tree. This construction is optimal with respect to the number of trees and maximum degree.

cs.CG↗

Charting the Diameter Computation Landscape of Geometric Intersection Graphs in Three Dimensions and Higher

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [SoCG '22] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter $Δ$ is at least $Ω(\log n)$, hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: - A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. - A truly subquadratic time lower bound for \Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be truly subquadratic hard when the diameter is $Ω(\log n)$. - A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D. - A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

cs.CG↗

Large induced acyclic and outerplanar subgraphs of 2-outerplanar graph

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related result, Chartran and Kronk, proved that the vertices of every planar graph can be partitioned into three sets, each of which induce a forest. We show tighter results for 2-outerplanar graphs. We show that every 2-outerplanar graph has an induced forest on at least half the vertices by showing that its vertices can be partitioned into two sets, each of which induces a forest. We also show that every 2-outerplanar graph has an induced outerplanar graph on at least two-thirds of its vertices, assuming that the connected components of the inner layer are two-connected.

math.CO↗

CodeTF: One-stop Transformer Library for State-of-the-art Code LLMs

Code intelligence plays a key role in transforming modern software engineering. Recently, deep learning-based models, especially Transformer-based large language models (LLMs), have demonstrated remarkable potential in tackling these tasks by leveraging massive open-source code data and programming language features. However, the development and deployment of such models often require expertise in both machine learning and software engineering, creating a barrier for the model adoption. In this paper, we present CodeTF, an open-source Transformer-based library for state-of-the-art Code LLMs and code intelligence. Following the principles of modular design and extensible framework, we design CodeTF with a unified interface to enable rapid access and development across different types of models, datasets and tasks. Our library supports a collection of pretrained Code LLM models and popular code benchmarks, including a standardized interface to train and serve code LLMs efficiently, and data features such as language-specific parsers and utility functions for extracting code attributes. In this paper, we describe the design principles, the architecture, key modules and components, and compare with other related library tools. Finally, we hope CodeTF is able to bridge the gap between machine learning/generative AI and software engineering, providing a comprehensive open-source solution for developers, researchers, and practitioners.

cs.SE↗

Tree-Like Shortcuttings of Trees

Sparse shortcuttings of trees -- equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter -- have been studied extensively (under different names and settings), since the pioneering works of [Yao82, Cha87, AS87, BTS94], initially motivated by applications to range queries, online tree product, and MST verification, to name a few. These constructions were also lifted from trees to other graph families using known low-distortion embedding results. The works of [Yao82, Cha87, AS87, BTS94] establish a tight tradeoff between hop-diameter and sparsity (or average degree) for tree shortcuttings and imply constant-hop shortcuttings for $n$-node trees with sparsity $O(\log^* n)$. Despite their small sparsity, all known constant-hop shortcuttings contain dense subgraphs (of sparsity $Ω(\log n)$), which is a significant drawback for many applications. We initiate a systematic study of constant-hop tree shortcuttings that are ``tree-like''. We focus on two well-studied graph parameters that measure how far a graph is from a tree: arboricity and treewidth. Our contribution is twofold. * New upper and lower bounds for tree-like shortcuttings of trees, including an optimal tradeoff between hop-diameter and treewidth for all hop-diameter up to $O(\log\log n)$. We also provide a lower bound for larger values of $k$, which together yield $\text{hop-diameter}\times \text{treewidth} = Ω((\log\log n)^2)$ for all values of hop-diameter, resolving an open question of [FL22, Le23]. [...]

cs.DS↗

Federated Domain Generalization with Latent Space Inversion

Federated domain generalization (FedDG) addresses distribution shifts among clients in a federated learning framework. FedDG methods aggregate the parameters of locally trained client models to form a global model that generalizes to unseen clients while preserving data privacy. While improving the generalization capability of the global model, many existing approaches in FedDG jeopardize privacy by sharing statistics of client data between themselves. Our solution addresses this problem by contributing new ways to perform local client training and model aggregation. To improve local client training, we enforce (domain) invariance across local models with the help of a novel technique, \textbf{latent space inversion}, which enables better client privacy. When clients are not \emph{i.i.d}, aggregating their local models may discard certain local adaptations. To overcome this, we propose an \textbf{important weight} aggregation strategy to prioritize parameters that significantly influence predictions of local models during aggregation. Our extensive experiments show that our approach achieves superior results over state-of-the-art methods with less communication overhead.

cs.LG↗

Approximating Euclidean Shallow-Light Trees

For a weighted graph $G = (V, E, w)$ and a designated source vertex $s \in V$, a spanning tree that simultaneously approximates a shortest-path tree w.r.t. source $s$ and a minimum spanning tree is called a shallow-light tree (SLT). Specifically, an $(α, β)$-SLT of $G$ w.r.t. $s \in V$ is a spanning tree of $G$ with root-stretch $α$ (preserving all distances between $s$ and the other vertices up to a factor of $α$) and lightness $β$ (its weight is at most $β$ times the weight of a minimum spanning tree of $G$). Despite the large body of work on SLTs, the basic question of whether a better approximation algorithm exists was left untouched to date, and this holds in any graph family. This paper makes a first nontrivial step towards this question by presenting two bicriteria approximation algorithms. For any $ε>0$, a set $P$ of $n$ points in constant-dimensional Euclidean space and a source $s\in P$, our first (respectively, second) algorithm returns, in $O(n \log n \cdot {\rm polylog}(1/ε))$ time, a non-Steiner (resp., Steiner) tree with root-stretch $1+O(ε\log ε^{-1})$ and weight at most $O(\mathrm{opt}_ε\cdot \log^2 ε^{-1})$ (resp., $O(\mathrm{opt}_ε\cdot \log ε^{-1})$), where $\mathrm{opt}_ε$ denotes the minimum weight of a non-Steiner (resp., Steiner) tree with root-stretch $1+ε$.

cs.CG↗

Towards a Unified Theory of Light Spanners I: Fast (Yet Optimal) Constructions

Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly. (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. (3) These constructions are optimal in the standard and crude sense, but not in a refined sense that takes into account a wider range of involved parameters. This work aims at addressing these shortcomings by presenting a unified framework of light spanners in a variety of graph classes. Informally, the framework boils down to a transformation from sparse spanners to light spanners; since the state-of-the-art for sparse spanners is much more advanced than that for light spanners, such a transformation is powerful. Our framework is developed in two papers. The current paper is the first of the two -- it lays the basis of the unified framework and then applies it to design fast constructions with optimal lightness for several graph classes. Among various applications and implications of our framework, we highlight here the following: _ In low-dimensional Euclidean spaces, we present an $O(n\log n)$-time construction of $(1+ε)$-spanners with lightness and degree both bounded by constants in the algebraic computation tree (ACT). Our construction resolves a major problem in the area of geometric spanners, which was open for three decades. _ In general graphs, for any $k \geq 2$, we construct a $(2k-1)(1+ε)$-spanner with lightness $O(n^{1/k})$ in $O(m α(m,n))$ time. This result for light spanners in general weighted graphs is surprising, as it outperforms the analog one for sparse spanners.

cs.CG↗

Uncertainty-Guided Checkpoint Selection for Reinforcement Finetuning of Large Language Models

Reinforcement learning (RL) finetuning is crucial to aligning large language models (LLMs), but the process is notoriously unstable and exhibits high variance across model checkpoints. In practice, selecting the best checkpoint is challenging: evaluating checkpoints on the validation set during training is computationally expensive and requires a good validation set, while relying on the final checkpoint provides no guarantee of good performance. We introduce an uncertainty-guided approach for checkpoint selection (UGCS) that avoids these pitfalls. Our method identifies hard question-answer pairs using per-sample uncertainty and ranks checkpoints by how well they handle these challenging cases. By averaging the rewards of the top-uncertain samples over a short training window, our method produces a stable and discriminative signal without additional forward passes or significant computation overhead. Experiments across three datasets and three LLMs demonstrate that it consistently identifies checkpoints with stronger generalization, outperforming traditional strategies such as relying on training or validation performance. These results highlight that models solving their hardest tasks with low uncertainty are the most reliable overall.

cs.LG↗

Probabilities Are All You Need: A Probability-Only Approach to Uncertainty Estimation in Large Language Models

Large Language Models (LLMs) exhibit strong performance across various natural language processing (NLP) tasks but remain vulnerable to hallucinations, generating factually incorrect or misleading outputs. Uncertainty estimation, often using predictive entropy estimation, is key to addressing this issue. However, existing methods often require multiple samples or extra computation to assess semantic entropy. This paper proposes an efficient, training-free uncertainty estimation method that approximates predictive entropy using the responses' top-$K$ probabilities. Moreover, we employ an adaptive mechanism to determine $K$ to enhance flexibility and filter out low-confidence probabilities. Experimental results on three free-form question-answering datasets across several LLMs demonstrate that our method outperforms expensive state-of-the-art baselines, contributing to the broader goal of enhancing LLM trustworthiness.

cs.LG↗

GRAD: Graph-Retrieved Adaptive Decoding for Hallucination Mitigation

Hallucination mitigation remains a persistent challenge for large language models (LLMs), even as model scales grow. Existing approaches often rely on external knowledge sources, such as structured databases or knowledge graphs, accessed through prompting or retrieval. However, prompt-based grounding is fragile and domain-sensitive, while symbolic knowledge integration incurs heavy retrieval and formatting costs. Motivated by knowledge graphs, we introduce Graph-Retrieved Adaptive Decoding (GRAD), a decoding-time method that grounds generation in corpus-derived evidence without retraining. GRAD constructs a sparse token transition graph by accumulating next-token logits across a small retrieved corpus in a single forward pass. During decoding, graph-retrieved logits are max-normalized and adaptively fused with model logits to favor high-evidence continuations while preserving fluency. Across three models and a range of question-answering benchmarks spanning intrinsic, extrinsic hallucination, and factuality tasks, GRAD consistently surpasses baselines, achieving up to 9.7$\%$ higher intrinsic accuracy, 8.6$\%$ lower hallucination rates, and 6.9$\%$ greater correctness compared to greedy decoding, while attaining the highest truth--informativeness product score among all methods. GRAD offers a lightweight, plug-and-play alternative to contrastive decoding and knowledge graph augmentation, demonstrating that statistical evidence from corpus-level token transitions can effectively steer generation toward more truthful and verifiable outputs.

cs.CL↗

DmC: Nearest Neighbor Guidance Diffusion Model for Offline Cross-domain Reinforcement Learning

Cross-domain offline reinforcement learning (RL) seeks to enhance sample efficiency in offline RL by utilizing additional offline source datasets. A key challenge is to identify and utilize source samples that are most relevant to the target domain. Existing approaches address this challenge by measuring domain gaps through domain classifiers, target transition dynamics modeling, or mutual information estimation using contrastive loss. However, these methods often require large target datasets, which is impractical in many real-world scenarios. In this work, we address cross-domain offline RL under a limited target data setting, identifying two primary challenges: (1) Dataset imbalance, which is caused by large source and small target datasets and leads to overfitting in neural network-based domain gap estimators, resulting in uninformative measurements; and (2) Partial domain overlap, where only a subset of the source data is closely aligned with the target domain. To overcome these issues, we propose DmC, a novel framework for cross-domain offline RL with limited target samples. Specifically, DmC utilizes $k$-nearest neighbor ($k$-NN) based estimation to measure domain proximity without neural network training, effectively mitigating overfitting. Then, by utilizing this domain proximity, we introduce a nearest-neighbor-guided diffusion model to generate additional source samples that are better aligned with the target domain, thus enhancing policy learning with more effective source samples. Through theoretical analysis and extensive experiments in diverse MuJoCo environments, we demonstrate that DmC significantly outperforms state-of-the-art cross-domain offline RL methods, achieving substantial performance gains.

cs.LG↗

Approximate Light Spanners in Planar Graphs

In their seminal paper, Althöfer et al. (DCG 1993) introduced the {\em greedy spanner} and showed that, for any weighted planar graph $G$, the weight of the greedy $(1+ε)$-spanner is at most $(1+\frac{2}ε) \cdot w(MST(G))$, where $w(MST(G))$ is the weight of a minimum spanning tree $MST(G)$ of $G$. This bound is optimal in an {\em existential sense}: there exist planar graphs $G$ for which any $(1+ε)$-spanner has a weight of at least $(1+\frac{2}ε) \cdot w(MST(G))$. However, as an {\em approximation algorithm}, even for a {\em bicriteria} approximation, the weight approximation factor of the greedy spanner is essentially as large as the existential bound: There exist planar graphs $G$ for which the greedy $(1+x ε)$-spanner (for any $1\leq x = O(ε^{-1/2})$) has a weight of $Ω(\frac{1}{ε\cdot x^2})\cdot w(G_{OPT, ε})$, where $G_{OPT, ε}$ is a $(1+ε)$-spanner of $G$ of minimum weight. Despite the flurry of works over the past three decades on approximation algorithms for spanners as well as on light(-weight) spanners, there is still no (possibly bicriteria) approximation algorithm for light spanners in weighted planar graphs that outperforms the existential bound. As our main contribution, we present a polynomial time algorithm for constructing, in any weighted planar graph $G$, a $(1+ε\cdot 2^{O(\log^* 1/ε)})$-spanner for $G$ of total weight $O(1)\cdot w(G_{OPT, ε})$. To achieve this result, we develop a new technique, which we refer to as {\em iterative planar pruning}. It iteratively modifies a spanner [...]

cs.DS↗

Truly Subquadratic Time Algorithms for Diameter and Related Problems in Graphs of Bounded VC-dimension

We give the first truly subquadratic time algorithm, with $O^*(n^{2-1/18})$ running time, for computing the diameter of an $n$-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance of a general framework, applicable to different graph families and distance problems. Surprisingly, our framework completely bypasses sublinear separators (or $r$-divisions) which were used in all previous algorithms. Instead, we use low-diameter decompositions in their most elementary form. We also exploit bounded VC-dimension of set systems associated with the input graph, as well as new ideas on geometric data structures. Among the numerous applications of the general framework, we obtain: 1. An $\tilde{O}(mn^{1-1/(2d)})$ time algorithm for computing the diameter of $m$-edge sparse unweighted graphs with constant VC-dimension $d$. The previously known algorithms by Ducoffe, Habib, and Viennot [SODA 2019] and Duraj, Konieczny, and Potȩpa [ESA 2024] are truly subquadratic only when the diameter is a small polynomial. Our result thus generalizes truly subquadratic time algorithms known for planar and minor-free graphs (in fact, it slightly improves the previous time bound for minor-free graphs). 2. An $\tilde{O}(n^{2-1/12})$ time algorithm for computing the diameter of intersection graphs of axis-aligned squares with arbitrary size. The best-known algorithm by Duraj, Konieczny, and Potȩpa [ESA 2024] only works for unit squares and is only truly subquadratic in the low-diameter regime. 3. The first algorithms with truly subquadratic complexity for other distance-related problems, including all-vertex eccentricities, Wiener index, and exact distance oracles. (... truncated to meet the arXiv abstract requirement.)

cs.DS↗

Optimal Padded Decomposition For Bounded Treewidth Graphs

A $(β,δ,Δ)$-padded decomposition of an edge-weighted graph $G = (V,E,w)$ is a stochastic decomposition into clusters of diameter at most $Δ$ such that for every vertex $v\in V$, the probability that $\rm{ball}_G(v,γΔ)$ is entirely contained in the cluster containing $v$ is at least $e^{-βγ}$ for every $γ\in [0,δ]$. Padded decompositions have been studied for decades and have found numerous applications, including metric embedding, multicommodity flow-cut gap, multicut, and zero extension problems, to name a few. In these applications, parameter $β$, called the padding parameter, is the most important parameter since it decides either the distortion or the approximation ratios. For general graphs with $n$ vertices, $β= Θ(\log n)$. Klein, Plotkin, and Rao showed that $K_r$-minor-free graphs have padding parameter $β= O(r^3)$, which is a significant improvement over general graphs when $r$ is a constant. A long-standing conjecture is to construct a padded decomposition for $K_r$-minor-free graphs with padding parameter $β= O(\log r)$. Despite decades of research, the best-known result is $β= O(r)$, even for graphs with treewidth at most $r$. In this work, we make significant progress toward the aforementioned conjecture by showing that graphs with treewidth $\rm{tw}$ admit a padded decomposition with padding parameter $O(\log \rm{tw})$, which is tight. As corollaries, we obtain an exponential improvement in dependency on treewidth in a host of algorithmic applications: $O(\sqrt{ \log n \cdot \log(\rm{tw})})$ flow-cut gap, max flow-min multicut ratio of $O(\log(\rm{tw}))$, an $O(\log(\rm{tw}))$ approximation for the 0-extension problem, an $\ell^{O(\log n)}_\infty$ embedding with distortion $O(\log \rm{tw})$, and an $O(\log \rm{tw})$ bound for integrality gap for the uniform sparsest cut.

cs.DS↗

Accelerating Long-Term Molecular Dynamics with Physics-Informed Time-Series Forecasting

Efficient molecular dynamics (MD) simulation is vital for understanding atomic-scale processes in materials science and biophysics. Traditional density functional theory (DFT) methods are computationally expensive, which limits the feasibility of long-term simulations. We propose a novel approach that formulates MD simulation as a time-series forecasting problem, enabling advanced forecasting models to predict atomic trajectories via displacements rather than absolute positions. We incorporate a physics-informed loss and inference mechanism based on DFT-parametrised pair-wise Morse potential functions that penalize unphysical atomic proximity to enforce physical plausibility. Our method consistently surpasses standard baselines in simulation accuracy across diverse materials. The results highlight the importance of incorporating physics knowledge to enhance the reliability and precision of atomic trajectory forecasting. Remarkably, it enables stable modeling of thousands of MD steps in minutes, offering a scalable alternative to costly DFT simulations.

cs.LG↗

CausalPlan: Empowering Efficient LLM Multi-Agent Collaboration Through Causality-Driven Planning

Large language model (LLM) agents-especially smaller, open-source models-often produce causally invalid or incoherent actions in collaborative tasks due to their reliance on surface-level correlations rather than grounded causal reasoning. This limitation undermines their performance in terms of coordination and planning in dynamic environments. We address this challenge with CausalPlan, a two-phase framework that integrates explicit structural causal reasoning into the LLM planning process. At the core of CausalPlan is the Structural Causal Action (SCA) model, which learns a causal graph from agent trajectories to capture how prior actions and current environment states influence future decisions. This structure is then used to guide action selection by assigning causal scores to LLM-generated proposals, reweighting them accordingly, or falling back to causally grounded alternatives when needed. By embedding this causal knowledge directly into the decision loop, CausalPlan constrains planning to intervention-consistent behaviours without requiring fine-tuning of the LLM itself. We evaluate CausalPlan on the Overcooked-AI benchmark across five multi-agent coordination tasks and four LLMs of varying sizes: Gemma-7B, Llama-8B, Qwen-14B, and Llama-70B. Experimental results show that CausalPlan consistently reduces invalid actions and improves collaboration in both AI-AI and human-AI settings, outperforming strong reinforcement learning baselines. Our findings highlight the value of causality-driven planning for deploying efficient, interpretable, and generalisable multi-agent LLM systems.

cs.AI↗