The paper has been discussed after the rebuttal that the reviewers found useful and actionable (e.g., concerns about the confidence bound). The paper is recommended for acceptance. All reviewers have acknowledged that the paper is well motivated, well written and establishes a nice interplay between theory and a practical problem of interest.

AI methods, such as generative models and reinforcement learning, have recently been applied to combinatorial optimization (CO) problems, especially NP-hard ones. This paper compares such GPU-based methods with classical CPU-based methods on Maximum Independent Set (MIS). Experiments on standard graph families show that AI-based algorithms fail to outperform and, in many cases, to match the solution quality of the state-of-art classical solver KaMIS running on a single CPU. Some GPU-based methods even perform similarly to the simplest heuristic, degree-based greedy. Even with post-processing techniques like local search, AI-based methods still perform worse than CPU-based solvers. We develop a new mode of analysis to reveal that non-backtracking AI methods, e.g. LTFT (which is based on GFlowNets), end up reasoning similarly to the simplest degree-based greedy approach, and thus worse than KaMIS. We also find that CPU-based algorithms, notably KaMIS, have strong performance on sparse random graphs, which appears to refute a well-known conjectured upper bound for efficient algorithms from Coja-Oghlan & Efthymiou (2015).

We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle harder problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with other additions, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads (IMO) 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that combines multiple search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry2 to 84% for $\textit{all}$ geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Last but not least, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems directly from natural language input.

End-effector trajectory tracking algorithms find joint motions that drive robot manipulators to track reference trajectories. In practical scenarios, anytime algorithms are preferred for their ability to quickly generate initial motions and continuously refine them over time. In this paper, we present an algorithmic framework that adapts common graph-based trajectory tracking algorithms to be anytime and enhances their efficiency and effectiveness. Our key insight is to identify guide paths that approximately track the reference trajectory and strategically bias sampling toward the guide paths. We demonstrate the effectiveness of the proposed framework by restructuring two existing graph-based trajectory tracking algorithms and evaluating the updated algorithms in three experiments.

This study explores the integration of Blackout Diffusion into the DIFUSCO framework for combinatorial optimization, specifically targeting the Traveling Salesman Problem (TSP). Inspired by the success of discrete-time diffusion models (D3PM) in maintaining structural integrity, we extend the paradigm to a continuous-time framework, leveraging the unique properties of Blackout Diffusion. Continuous-time modeling introduces smoother transitions and refined control, hypothesizing enhanced solution quality over traditional discrete methods. We propose three key improvements to enhance the diffusion process. First, we transition from a discrete-time-based model to a continuous-time framework, providing a more refined and flexible formulation. Second, we refine the observation time scheduling to ensure a smooth and linear transformation throughout the diffusion process, allowing for a more natural progression of states. Finally, building upon the second improvement, we further enhance the reverse process by introducing finer time slices in regions that are particularly challenging for the model, thereby improving accuracy and stability in the reconstruction phase. Although the experimental results did not exceed the baseline performance, they demonstrate the effectiveness of these methods in balancing simplicity and complexity, offering new insights into diffusion-based combinatorial optimization. This work represents the first application of Blackout Diffusion to combinatorial optimization, providing a foundation for further advancements in this domain. * The code is available for review at https://github.com/Giventicket/BlackoutDIFUSCO.

Recent advancements in large language models (LLMs) have spurred growing interest in automatic theorem proving using Lean4, where effective tree search methods are crucial for navigating proof search spaces. While the existing approaches primarily rely on value functions and Monte Carlo Tree Search (MCTS), the potential of simpler methods like Best-First Search (BFS) remains underexplored. This paper investigates whether BFS can achieve competitive performance in large-scale theorem proving tasks. We present \texttt{BFS-Prover}, a scalable expert iteration framework, featuring three key innovations. First, we implement strategic data filtering at each expert iteration round, excluding problems solvable via beam search node expansion to focus on harder cases. Second, we improve the sample efficiency of BFS through Direct Preference Optimization (DPO) applied to state-tactic pairs automatically annotated with compiler error feedback, refining the LLM's policy to prioritize productive expansions. Third, we employ length normalization in BFS to encourage exploration of deeper proof paths. \texttt{BFS-Prover} achieves a score of $71.31$ on the MiniF2F test set and therefore challenges the perceived necessity of complex tree search methods, demonstrating that BFS can achieve competitive performance when properly scaled.

Diffusion models have recently advanced Combinatorial Optimization (CO) as a powerful backbone for neural solvers. However, their iterative sampling process requiring denoising across multiple noise levels incurs substantial overhead. We propose to learn direct mappings from different noise levels to the optimal solution for a given instance, facilitating high-quality generation with minimal shots. This is achieved through an optimization consistency training protocol, which, for a given instance, minimizes the difference among samples originating from varying generative trajectories and time steps relative to the optimal solution. The proposed model enables fast single-step solution generation while retaining the option of multi-step sampling to trade for sampling quality, which offers a more effective and efficient alternative backbone for neural solvers. In addition, within the training-to-testing (T2T) framework, to bridge the gap between training on historical instances and solving new instances, we introduce a novel consistency-based gradient search scheme during the test stage, enabling more effective exploration of the solution space learned during training. It is achieved by updating the latent solution probabilities under objective gradient guidance during the alternation of noise injection and denoising steps. We refer to this model as Fast T2T. Extensive experiments on two popular tasks, the Traveling Salesman Problem (TSP) and Maximal Independent Set (MIS), demonstrate the superiority of Fast T2T regarding both solution quality and efficiency, even outperforming LKH given limited time budgets. Notably, Fast T2T with merely one-step generation and one-step gradient search can mostly outperform the SOTA diffusion-based counterparts that require hundreds of steps, while achieving tens of times speedup.

Despite their remarkable capabilities, large language models often struggle with tasks requiring complex reasoning and planning. While existing approaches like Chain-of-Thought prompting and tree search techniques show promise, they are limited by their reliance on predefined heuristics and computationally expensive exploration strategies. We propose Policy-Guided Tree Search (PGTS), a framework that combines reinforcement learning with structured tree exploration to efficiently navigate reasoning paths. Our key innovation is a learned policy that dynamically decides between expanding, branching, backtracking, or terminating exploration, eliminating the need for manual heuristics or exhaustive search. Experiments across mathematical reasoning, logical deduction, and planning benchmarks demonstrate that PGTS achieves superior reasoning performance while significantly reducing computational costs compared to existing methods. These results establish PGTS as a scalable and effective solution for tackling complex reasoning tasks with LLMs.

In mixture models, nonspherical (anisotropic) noise within each cluster is widely present in real-world data. We study both the minimax rate and optimal statistical procedure for clustering under high-dimensional nonspherical mixture models. In high-dimensional settings, we first establish the information-theoretic limits for clustering under Gaussian mixtures. The minimax lower bound unveils an intriguing informational dimension-reduction phenomenon: there exists a substantial gap between the minimax rate and the oracle clustering risk, with the former determined solely by the projected centers and projected covariance matrices in a low-dimensional space. Motivated by the lower bound, we propose a novel computationally efficient clustering method: Covariance Projected Spectral Clustering (COPO). Its key step is to project the high-dimensional data onto the low-dimensional space spanned by the cluster centers and then use the projected covariance matrices in this space to enhance clustering. We establish tight algorithmic upper bounds for COPO, both for Gaussian noise with flexible covariance and general noise with local dependence. Our theory indicates the minimax-optimality of COPO in the Gaussian case and highlights its adaptivity to a broad spectrum of dependent noise. Extensive simulation studies under various noise structures and real data analysis demonstrate our method's superior performance.

These tools apply basic strategies that work well for random problems but miss critical patterns in structured instances. SAT encodings of real problems contain inherited patterns from graph layouts, data connections, and domain-specific rules. The transformation to Conjunctive Normal Form (CNF) obscures these patterns. Current local search methods skip these structures in favor of general approaches. This paper addresses these limitations by introducing a framework that leverages LLMs to generate local search strategies tailored to encoding structures, enabling solvers to take advantage of these patterns for improved performance. Our research addresses three questions: 1. How can LLMs analyze PySAT [Ignatiev et al., 2024] code to interpret how problem structure translates to SAT clauses? 2. How can we create local search strategies that recognize and exploit these encoding patterns?