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Deep Learning · Large Language Models

Nian Ran, Zhongzheng Li, Yue Wang, Qingsong Ran, Xiaoyuan Zhang, Shikun Feng, Richard Allmendinger, Xiaoguang Zhao

Multi-objective discrete optimization problems, such as molecular design, pose significant challenges due to their vast and unstructured combinatorial spaces. Traditional evolutionary algorithms often get trapped in local optima, while expert knowledge can provide crucial guidance for accelerating convergence. Large language models (LLMs) offer powerful priors and reasoning ability, making them natural optimizers when expert knowledge matters. However, closed-source LLMs, though strong in exploration, cannot update their parameters and thus cannot internalize experience. Conversely, smaller open models can be continually fine-tuned but lack broad knowledge and reasoning strength. We introduce Multi-LLM Collaborative Co-evolution (MCCE), a hybrid framework that unites a frozen closed-source LLM with a lightweight trainable model. The system maintains a trajectory memory of past search processes; the small model is progressively refined via reinforcement learning, with the two models jointly supporting and complementing each other in global exploration. Unlike model distillation, this process enhances the capabilities of both models through mutual inspiration. Experiments on multi-objective drug design benchmarks show that MCCE achieves state-of-the-art Pareto front quality and consistently outperforms baselines. These results highlight a new paradigm for enabling continual evolution in hybrid LLM systems, combining knowledge-driven exploration with experience-driven learning.The code of MCCE is available on https://anonymous.4open.science/r/MCCE_Anonymous-1F92

Optimization · Discrete and Combinatorial Optimization

Peng Chen, Hailiang Zhao, Xueyan Tang, Yixuan Wang, Shuiguang Deng

Learning-augmented paging has been extensively studied in recent years. A key advantage over naive ML-based approaches is \emph{bounded robustness}, which guarantees worst-case performance even when predictions are inaccurate, making these algorithms valuable for real-world systems. Prior work achieves robustness bounds of $2H_k + O(1)$ in the randomized setting, leaving a gap to the optimal competitive ratio $H_k$. We are the first to study how to close this gap. In this paper, we begin by analyzing online optimality and provide a new proof of the latest $H_k$-competitive algorithm, which facilitates analysis in the learning-augmented setting. Then, we review existing learning-augmented paging algorithms and introduce a unifying primitive, the \emph{relative prediction budget}, which captures the essence of how to establish robustness and reveals that prior algorithms either overuse or underutilize predictions. Guided by the above analysis, we develop a new framework that achieves the best-possible robustness for learning-augmented paging: $H_k + O(1)$. Experiments further demonstrate strong practical performance.

Optimization · Discrete and Combinatorial Optimization

Shengyu Feng, Tarun Suresh, Yiming Yang

Neural solvers have recently emerged as powerful tools for combinatorial optimization (CO). Among them, diffusion models have shown strong promise due to their ability to capture highly multimodal solution distributions in CO through iterative generative processes. However, training diffusion models typically requires large collections of near-optimal solutions, which limits their scalability and generalization. We address this fundamental challenge by extending adjoint matching, a powerful unsupervised diffusion training framework based on chain-rule–style gradient propagation in continuous spaces, to discrete combinatorial domains. Our approach resolves the broken-gradient issue inherent to discrete data and unifies local and global training objectives within a single principled framework. Empirically, our method consistently outperforms existing unsupervised baselines and achieves performance comparable to supervised diffusion models.

Optimization · Discrete and Combinatorial Optimization

Xinyi Ke, Kai Li, Junliang Xing, Yifan Zhang, Jian Cheng

Large language models (LLMs) have enabled rapid progress in automatic heuristic discovery (AHD), yet most existing methods are predominantly limited by static evaluation against fixed instance distributions, leading to potential overfitting and poor generalization under distributional shifts. We propose Algorithm Space Response Oracles (ASRO), a game-theoretic framework that reframes heuristic discovery as a program level co-evolution between solver and instance generator. ASRO models their interaction as a two-player zero-sum game, maintains growing strategy pools on both sides, and iteratively expands them via LLM-based best-response oracles against mixed opponent meta-strategies, thereby replacing static evaluation with an adaptive, self-generated curriculum. Across multiple combinatorial optimization domains, ASRO consistently outperforms static-training AHD baselines built on the same program search mechanisms, achieving substantially improved generalization and robustness on diverse and out-of-distribution instances.

Optimization · Discrete and Combinatorial Optimization

Achref Jaziri, Thibaut Cuvelier, Bruno De Backer

Large-scale Set Cover Problems (SCP) with millions of variables and complex cost structures require high-quality solutions within seconds, yet remain beyond the reach of exact solvers and pose severe generalization challenges for neural methods. Such problems necessitate decomposition into bounded subproblems; however, when the induced subproblem topology differs from that observed during training, existing neural approaches often fail to transfer reliably. We introduce Generative Large Neighborhood Search (GLNS), which reframes neighborhood selection as generation using a discrete diffusion model. Our key insight is that the diffusion denoising trajectory exposes variables exhibiting high prediction instability across timesteps and identifies regions where local repair yields downstream improvement. GLNS exploits this trajectory-level signal to construct high-impact neighborhoods via a localized, bounded-complexity generative sampling procedure, enabling robust neighborhood selection without retraining. As a result, GLNS transfers effectively across cost regimes and instance scales within SCP. Under tight and equal wall-clock budgets, GLNS consistently outperforms established neural baselines and achieves competitive performance with state-of-the-art MIP solvers. These results demonstrate trajectory-guided generation as a scalable framework for large-scale SCP and suggest potential relevance to other constrained optimization settings.

Optimization · Discrete and Combinatorial Optimization

Yixin Huang, Shengyu Feng, Yiming Yang

Existing neural combinatorial optimization (CO) solvers often rely heavily on expensive labeled data and additional post-processing to produce feasible solutions. Research into mixed integer linear programs (MILPs) is particularly limited due to the lack of effective heuristics for feasibility and the challenge of modeling mixed-type variables for neural solvers. To address these issues, we propose a novel unsupervised Langevin sampler for solving MILPs. Our framework learns only integer variables, while continuous variables are solved using an exact linear programming solver, thus isolating the combinatorial hardness of the problem and avoiding unnecessary modeling complexity. The sampler is based on Langevin dynamics and incorporates both objective optimization and constraint satisfaction into a unified energy function, enabling the model to jointly learn feasibility and optimality. Experiments demonstrate that our method achieves 100\% feasibility without expensive post-processing and matches or outperforms supervised solvers on benchmark datasets, highlighting its effectiveness and scalability.

Optimization · Discrete and Combinatorial Optimization

Hongyu Cheng, Amitabh Basu

Machine learning is increasingly used to guide branch-and-cut (B&C) for mixed-integer linear programming by learning score-based policies for selecting branching variables and cutting planes. Many approaches train on local signals from lookahead heuristics such as strong branching, and linear programming (LP) bound improvement for cut selection. Training and evaluation of the learned models often focus on local score accuracy. We show that such local score-based methods can lead to search trees exponentially larger than optimal tree sizes, by identifying two sources of this gap. The first is that these widely used expert signals can be misaligned with overall tree size. LP bound improvement can select a root cut set that yields an exponentially larger strong branching tree than selecting cuts by a simple proxy score, and strong branching itself can be exponentially suboptimal (Dey et al., 2024). The second is that small discrepancies can be amplified by the branch-and-bound recursion. An arbitrarily small perturbation of the right-hand sides in a root cut set can change the minimum tree size from a single node to exponentially many. For branching, arbitrarily small score discrepancies, and differences only in tie-breaking, can produce trees of exponentially different sizes, and even a small number of decision differences along a trajectory can incur exponential growth. These results show that branch-and-cut policies trained and learned using local expert scores do not guarantee small trees, thus motivating the study of data-driven methods that produce policies better aligned with tree size rather than only accuracy on expert scores.

Optimization · Discrete and Combinatorial Optimization

Zubin Zheng, Jiahao Wu, Shengcai Liu

The quantum approximate optimization algorithm (QAOA) holds promise for combinatorial optimization but is constrained by limited qubits. While divide-and-conquer frameworks like QAOA$^2$ address scalability by partitioning graphs into subgraphs, existing methods suffer from two fundamental limitations: i) misalignment between heuristic partitioning metrics and quantum optimization goals, and ii) topology-blind parameter initialization that leads to optimization cold starts. To bridge these gaps, we propose **Neural QAOA$^2$**, an end-to-end differentiable framework that jointly generates graph partitions and initial parameters. By integrating a generative evaluative network (GEN), our method utilizes a differentiable quantum evaluator as a high-fidelity performance surrogate to provide direct gradient guidance, enabling the joint generator to learn the intrinsic mapping from graph topology to high-quality partition and parameter configurations. Extensive experiments on 183 QUBO, Ising, and MaxCut instances (21 to 1000 variables) demonstrate that our gradient-driven approach significantly outperforms heuristic baselines, ranking first on 101 instances. It exhibits zero-shot generalization across out-of-distribution graph topologies and scales.

Optimization · Discrete and Combinatorial Optimization

Changliang Zhou, Canhong Yu, Shunyu Yao, Xi Lin, Zhenkun Wang, Yu Zhou, Qingfu Zhang

Multi-task neural routing solvers have emerged as a promising paradigm for their ability to solve multiple vehicle routing problems (VRPs) using a single model. However, existing neural solvers typically rely on predefined problem constraints or require per-problem fine-tuning, which substantially limits their zero-shot generalization ability to unseen VRP variants. To address this critical bottleneck, we propose URS, a unified neural routing solver that achieves zero-shot generalization across a wide range of unseen VRPs with a single model. We propose a unified data representation (UDR) that replaces problem enumeration with data unification, thereby broadening the problem coverage and reducing reliance on domain expertise. In addition, we introduce a Mixed Bias Module (MBM) during encoding to improve node embeddings, which efficiently captures multiple priors inherent to various problems. On top of the UDR, we develop a problem-conditioned parameter generator to further improve zero-shot generalization. Extensive experiments show that URS consistently produces high-quality solutions for 110 VRP variants (including 99 unseen variants) while demonstrating impressive scalability to large-scale instances with up to 7000 nodes. To the best of our knowledge, URS is the first neural solver to handle over 100 VRP variants with a single model.

Deep Learning · Graph Neural Networks

Yumeng Zhao, Huiying Hu, Steve Wen, Junjie Shen, Bei Hua

Graph Neural Networks (GNNs) are increasingly adopting the "pre-training, adaptation" paradigm, which first pre-train GNNs on large-scale unlabeled graph data and then adapt them to specific downstream tasks. As a common pattern, graph prompting adapts to the frozen encoder by modifying the input graph structure, rather than fine-tuning the model parameters. However, most existing graph prompting approaches optimize the continuous and weighted adjacency structure in the adaptation phase, while requiring a hard discretization at inference time. This difference causes a train-test mismatch which is particularly harmful in few-shot regimes. To address the issue, we propose **Di**screte **P**rompting for **G**raphs, a discrete prompting framework that directly learns task-specific topology prompts in the combinatorial space. DiP-G operates on multi-hop local candidate subgraphs to ensure scalability, generates hard (k)-sparse prompts through a perturbed Top-(k) solver, and optimizes the discrete structures using an I-MLE gradient estimator. To improve the efficiency of backward pass, we further introduce an adaptive active-set screening rule that accelerates the target solve and can provably maintain the accuracy of the solution. Extensive experiments conducted on multiple benchmark datasets have validated the effectiveness of our proposed method. Our main code is available in the supplementary materials.

Optimization · Discrete and Combinatorial Optimization

Debraj Banerjee, Santanu Mahapatra, Kunal Narayan Chaudhury

The generalized Ising problem captures a broad spectrum of hard combinatorial problems, including MAX-CUT, Number Partitioning (NPP), and Maximum Independent Set. In this work, we consider the notion of one-flip local minima for this problem. We construct a polynomial relaxation and prove that there exists a one-to-one correspondence between the local minima of the relaxation and the one-flip minima of the original Ising problem. This guarantee reduces the Ising problem to finding the local minima of a smooth function, allowing us to leverage scalable gradient-based optimizers such as ADAM. We demonstrate that our method achieves strong performance across challenging benchmarks, including spin-glass models, MAX-CUT, and NPP.

Optimization · Discrete and Combinatorial Optimization

Yang Huang, Yifan Zhang, Jian Cheng

We propose an OT-guided sampling framework for solving graph combinatorial optimization through exact multilinear relaxation. Graph combinatorial optimization problems can be written as quadratic unconstrained binary optimization(QUBO). Leveraging a classical result in combinatorial optimization, we obtain a continuous multi-linear relaxation of QUBO that is exact, in the sense that it preserves the optimal binary solutions. The challenge is that the resulting energy landscape is highly nonconvex. We address this by treating the objective as an energy function and optimizing via sampling from the induced Boltzmann distribution to escape poor local optima. Viewing sampling as transporting a simple reference distribution to the target distribution, we use optimal transport to characterize more efficient probability flow and derive a stochastic optimal control problem whose solution yields an optimal sampling dynamics. We parameterize the control policy with graph neural networks to approximate the optimal control. Experiments show improved solution quality and efficiency over strong combinatorial and learning-based baselines.

Optimization · Large Scale, Parallel and Distributed

Jingyi Zhao, Linxin Yang, Haohua Zhang, Qile He, Tian Ding

Dynamic programming (DP) is central to combinatorial optimization, optimal control, and reinforcement learning, yet its perceived sequentiality has long hindered scalability. We introduce a general-purpose GPU framework that reformulates broad classes of forward DP recursions as batched min--plus matrix--vector products over layered DAGs, collapsing actions into masked state-to-state transitions that map directly to GPU kernels. This approach removes a major bottleneck in scenario-based stochastic programming (SP), where the use of DP has traditionally restricted the number of scenarios due to excessive computational cost. Our framework exposes massive parallelism across scenarios, transition layers, and, when applicable, route or action options, via self-designed GPU kernels that implement Bellman updates with warp-/block-level reductions and numerically safe masking. In a single GPU pass, these kernels can process over $10^6$ uncertainty realizations, far beyond the capacity of prior scenario-based methods. We demonstrate the approach in two canonical SP applications: (i) a vectorized split operator for the capacitated vehicle routing problem with stochastic demand, exploiting **2D** parallelism (scenarios $\times$ transitions); and (ii) a forward inventory reinsertion DP under an order-up-to policy, exploiting **3D** parallelism (scenarios $\times$ inventory transitions $\times$ route options). Across benchmarks, the implementation scales nearly linearly in the number of scenarios and achieves one to three orders of magnitude speedups over multithreaded CPU baselines, yielding tighter SAA estimates and consistently stronger first-stage decisions under identical wall-clock budgets. Viewed as hardware-aware software primitives, our min--plus DP kernels offer a drop-in path to scalable, GPU-accelerated stochastic discrete optimization.

Applications · Chemistry and Drug Discovery

James Bowden, Sergey Levine, Jennifer Listgarten

In the era of AI-driven science and engineering, we often want to design discrete objects (e.g., circuits, proteins, materials) in silico according to user-specified properties (e.g., that a protein binds its target). Given a property predictive model, in silico design typically involves training a generative model over the design space (e.g., over the set of all length-L proteins) to concentrate on designs with the desired properties. Distributional optimization, formalized as an estimation of distribution algorithm or as reinforcement learning policy optimization, maximizes an objective function in expectation over samples. Optimizing a distribution over discrete-valued designs is in general challenging due to the combinatorial nature of the design space. However, many property predictors in scientific applications are decomposable in the sense that they can be factorized over design variables in a way that will prove useful. For example, the active site amino acids in a catalytic protein may need to only loosely interact with the rest of the protein for maximal catalytic activity. Current distributional optimization algorithms are unable to make use of such structure, which could dramatically improve the optimization. Herein, we propose and demonstrate use of a new distributional optimization algorithm, Decomposition-Aware Distributional Optimization (DADO), that can leverage any decomposability defined by a junction tree on the design variables. At its core, DADO employs a factorized “search distribution”—a learned generative model—for efficient navigation of the search space, and invokes graph message passing to coordinate optimization across all variables.

Seul Lee, Karsten Kreis, Srimukh Veccham, Meng Liu, Danny Reidenbach, Saee Paliwal, Weili Nie, Arash Vahdat

A well-known pitfall of molecular generative models is that they are not guaranteed to generate synthesizable molecules. Existing solutions for this problem often struggle to effectively navigate exponentially large combinatorial space of synthesizable molecules and suffer from poor coverage. To address this problem, we introduce ReaSyn, an iterative generative pathway refinement framework that obtains synthesizable analogs to input molecules by projecting them onto synthesizable space. Specifically, we propose a simple synthetic pathway representation that allows for generating pathways in both bottom-up and top-down traversal of synthetic trees. We design ReaSyn so that both bottom-up and top-down pathways can be sampled with a single unified autoregressive model. ReaSyn can thus iteratively refine subtrees of generated synthetic trees in a bidirectional manner. Further, we introduce a discrete flow model that refines the generated pathway at the entire pathway level with edit operations: insertion, deletion, and substitution. The iterative refinement cycle of (1) bottom-up decoding, (2) top-down decoding, and (3) holistic editing constitutes a powerful pathway reasoning strategy, allowing the model to explore the vast space of synthesizable molecules. Experimentally, ReaSyn achieves the highest reconstruction rate and pathway diversity in synthesizable molecule reconstruction and the highest optimization performance in synthesizable goal-directed molecular optimization, and significantly outperforms previous synthesizable projection methods in synthesizable hit expansion. These results highlight ReaSyn's superior ability to navigate combinatorially-large synthesizable chemical space.

Optimization · Non-Convex

Peng Xu, Chun Ying Hou, Xiaohui Chen, Richard Zhang

Clustering is a hard discrete optimization problem. Nonconvex approaches such as low-rank semidefinite programming (SDP) have recently demonstrated promising statistical and local algorithmic guarantees for cluster recovery. Due to the combinatorial structure of the $K$-means clustering problem, current relaxation algorithms struggle to balance their constraint feasibility and objective optimality, presenting tremendous challenges in computing the second-order critical points with rigorous guarantees. In this paper, we provide a new formulation of the $K$-means problem as a smooth unconstrained optimization over a submanifold and characterize its Riemannian structures to allow it to be solved using a second-order cubic-regularized Riemannian Newton algorithm. By factorizing the $K$-means manifold into a product manifold, we show how each Newton subproblem can be solved in linear time. Our numerical experiments show that the proposed method converges significantly faster than the state-of-the-art first-order nonnegative low-rank factorization method, while achieving similarly optimal statistical accuracy.

Deep Learning · Generative Models and Autoencoders

Alperen Gozeten, Muhammed Ildiz, Xuechen Zhang, Hrayr Harutyunyan, Ankit Singh Rawat, Samet Oymak

Modern language models generate chain-of-thought traces by autoregressively sampling tokens from a finite vocabulary. While this discrete sampling has achieved remarkable success, conducting chain-of-thought with continuously-valued tokens (CoT2) offers a richer and more expressive alternative. Our work provides new theoretical guarantees and algorithms for CoT2, motivated by logical reasoning tasks that inherently require search capabilities. Theoretically, we establish how CoT2 facilitates the model to track multiple discrete traces in parallel; and quantify the level of achievable parallelism and its benefits for inference efficiency. We also provide a CoT2-based one-layer transformer construction that solves the combinatorial ``subset sum problem'' given a sufficient embedding dimension. These insights arise from a novel and effective supervision strategy where we match the language model outputs to the empirical token distributions of a set of target traces. Complementing this, we introduce sampling strategies that unlock policy optimization methods for CoT2. Our primary strategy samples and composes $K$ discrete tokens at each decoding step to control the level of parallelism. Experiments confirm that (i) the optimal level of parallelism is governed by the embedding dimension, (ii) our continuous supervision strategy can outperform alternative methods, and (iii) policy optimization with CoT2 indeed improves the performance of the model beyond its initial discrete or continuous supervision.

Applications · Chemistry and Drug Discovery

Seul Lee, Karsten Kreis, Srimukh Veccham, Meng Liu, Danny Reidenbach, Saee Paliwal, Weili Nie, Arash Vahdat

A well-known pitfall of molecular generative models is that they are not guaranteed to generate synthesizable molecules. Existing solutions for this problem often struggle to effectively navigate exponentially large combinatorial space of synthesizable molecules and suffer from poor coverage. To address this problem, we introduce ReaSyn, an iterative generative pathway refinement framework that obtains synthesizable analogs to input molecules by projecting them onto synthesizable space. Specifically, we propose a simple synthetic pathway representation that allows for generating pathways in both bottom-up and top-down traversal of synthetic trees. We design ReaSyn so that both bottom-up and top-down pathways can be sampled with a single unified autoregressive model. ReaSyn can thus iteratively refine subtrees of generated synthetic trees in a bidirectional manner. Further, we introduce a discrete flow model that refines the generated pathway at the entire pathway level with edit operations: insertion, deletion, and substitution. The iterative refinement cycle of (1) bottom-up decoding, (2) top-down decoding, and (3) holistic editing constitutes a powerful pathway reasoning strategy, allowing the model to explore the vast space of synthesizable molecules. Experimentally, ReaSyn achieves the highest reconstruction rate and pathway diversity in synthesizable molecule reconstruction and the highest optimization performance in synthesizable goal-directed molecular optimization, and significantly outperforms previous synthesizable projection methods in synthesizable hit expansion. These results highlight ReaSyn's superior ability to navigate combinatorially-large synthesizable chemical space.

Jiahao Zhang, Anoop Cherian, Cristian Rodriguez, Weijian Deng, Stephen Gould

Assembling furniture amounts to solving the discrete-continuous optimization task of selecting the furniture parts to assemble and estimating their connecting poses in a physically realistic manner. The problem is hampered by its combinatorially large yet sparse solution space thus making learning to assemble a challenging task for current machine learning models. In this paper, we attempt to solve this task by leveraging the assembly instructions provided in diagrammatic manuals that typically accompany the furniture parts. Our key insight is to use the cues in these diagrams to split the problem into discrete and continuous phases. Specifically, we present Manual-PA, a transformer-based instruction Manual-guided 3D Part Assembly framework that learns to semantically align 3D parts with their illustrations in the manuals using a contrastive learning backbone towards predicting the assembly order and infers the 6D pose of each part via relating it to the final furniture depicted in the manual. To validate the efficacy of our method, we conduct experiments on the benchmark PartNet dataset. Our results show that using the diagrams and the order of the parts lead to significant improvements in assembly performance against the state of the art. Further, Manual-PA demonstrates strong generalization to real-world IKEA furniture assembly on the IKEA-Manual dataset.

Niloy Talukder, Croix Gyurek, Mohammad Al Hasan

With the adoption of deep learning models to low-power, small-memory edge devices, energy consumption and storage usage of such models have become a key concern. The problem exacerbates even further with ever-growing data and equally-matched bulkier models. This concern is particularly pronounced for graph data due to its quadratic storage, irregular (non-grid) geometry, and very large size. Typical graph data, such as road networks, infrastructure networks, and social networks, easily exceeds millions of nodes, and several gigabytes of storage is needed just to store the node embedding vectors, let alone the model parameters. In recent years, the memory issue has been addressed by moving away from memory-intensive double precision floating-point arithmetic towards single-precision or even half-precision, often by trading-off marginally small performance. Along this effort, we propose Node2Binary, which embeds graph nodes in as few as 128 binary bits, thereby reducing the memory footprint of vertex embedding vectors by several orders of magnitude. Node2Binary. leverages a fast community detection algorithm to convert the given graph into a hierarchical partition tree and then find embeddings of graph vertices in binary space by solving a combinatorial optimization (CO) task over the tree edges. CO is NP-hard, but Node2Binary uses an innovative combination of discrete gradient descent and randomization to solve this task effectively and efficiently. Extensive experiments over four real-world graphs show that Node2Binary achieves competitive performance compared to the state-of-the art graph embedding methods in both node classification and link prediction tasks.