Constraint Programming (CP) offers an intuitive, declarative framework for modeling V ehicle Routing Problems (VRP), yet classical CP models based on successor variables cannot always deal with optional visits or insertion based heuristics. T o address these limitations, this paper formalizes sequence variables within CP . Unlike the classical successor models, this computational domain handle optional visits and support insertion heuristics, including insertion-based Large Neighborhood Search. We provide a clear definition of their domain, update operations, and introduce consistency levels for constraints on this domain. An implementation is described with the underlying data structures required for integrating sequence variables into existing trail-based CP solvers. Furthermore, global constraints specifically designed for sequence variables and vehicle routing are introduced. Finally, the effectiveness of sequence variables is demonstrated by simplifying problem modeling and achieving competitive computational performance on the Dial-a-Ride Problem.

We introduce Constraint-Aware Federated Learning with Lagrangian Dual Optimization (CAFL-L), a principled extension of FedAvg that explicitly incorporates device-level resource constraints including energy, communication, memory, and thermal budgets. CAFL-L employs Lagrangian dual optimization to dynamically adapt training hyperparameters -- freezing depth, local steps, batch size, and communication compression -- while preserving training stability through token-budget preservation via gradient accumulation. Experiments on a character-level language model demonstrate that CAFL-L achieves superior constraint satisfaction compared to standard FedAvg (reducing memory usage by 20% and communication by 95%) while maintaining competitive validation performance, making it practical for deployment on resource-constrained edge devices.

Centralized trajectory optimization in the joint space of multiple robots allows access to a larger feasible space that can result in smoother trajectories, especially while planning in tight spaces. Unfortunately, it is often computationally intractable beyond a very small swarm size. In this paper, we propose Flow-Opt, a learning-based approach towards improving the computational tractability of centralized multi-robot trajectory optimization. Specifically, we reduce the problem to first learning a generative model to sample different candidate trajectories and then using a learned Safety-Filter(SF) to ensure fast inference-time constraint satisfaction. We propose a flow-matching model with a diffusion transformer (DiT) augmented with permutation invariant robot position and map encoders as the generative model. We develop a custom solver for our SF and equip it with a neural network that predicts context-specific initialization. The initialization network is trained in a self-supervised manner, taking advantage of the differentiability of the SF solver. We advance the state-of-the-art in the following respects. First, we show that we can generate trajectories of tens of robots in cluttered environments in a few tens of milliseconds. This is several times faster than existing centralized optimization approaches. Moreover, our approach also generates smoother trajectories orders of magnitude faster than competing baselines based on diffusion models. Second, each component of our approach can be batched, allowing us to solve a few tens of problem instances in a fraction of a second. We believe this is a first such result; no existing approach provides such capabilities. Finally, our approach can generate a diverse set of trajectories between a given set of start and goal locations, which can capture different collision-avoidance behaviors.

Functional brain network (FBN) modeling often relies on local pairwise interactions, whose limitation in capturing high-order dependencies is theoretically analyzed in this paper. Meanwhile, the computational burden and heuristic nature of current hypergraph modeling approaches hinder end-to-end learning of FBN structures directly from data distributions. To address this, we propose to extract high-order FBN structures under global constraints, and implement this as a Global Constraints oriented Multi-resolution (GCM) FBN structure learning framework. It incorporates 4 types of global constraint (signal synchronization, subject identity, expected edge numbers, and data labels) to enable learning FBN structures for 4 distinct levels (sample/subject/group/project) of modeling resolution. Experimental results demonstrate that GCM achieves up to a 30.6% improvement in relative accuracy and a 96.3% reduction in computational time across 5 datasets and 2 task settings, compared to 9 baselines and 10 state-of-the-art methods. Extensive experiments validate the contributions of individual components and highlight the interpretability of GCM. This work offers a novel perspective on FBN structure learning and provides a foundation for interdisciplinary applications in cognitive neuroscience. Code is publicly available on https://github.com/lzhan94swu/GCM.

While researchers have made significant progress in enabling large language models (LLMs) to perform multi-step planning, LLMs struggle to ensure that those plans align with high-level user intent and satisfy symbolic constraints, especially in complex, multi-step domains. Existing reasoning approaches such as Chain-of-Thought (CoT), Tree-of-Thought (ToT), and verifier-augmented methods, expand the search space but often yield infeasible actions or hallucinated steps. To overcome these limitations, we propose Constraints-of-Thought (Const-o-T), a framework that provides a structured prior that enables Monte Carlo Tree Search (MCTS) focus search on semantically meaningful paths. Each reasoning step is represented as an (intent, constraint) pair, which serves both to compress the search space and enforce validity. Unlike prior methods that merely generate reasoning traces or validate outputs post hoc, Const-o-T uses (intent, constraint)pairs to actively focus the search toward feasible and meaningful plans. We integrate Const-o-T into MCTS using a structured representation of intent-constraint pairs constraints prune infeasible branches and guide exploration toward semantically valid actions, improving planning efficiency and verifiable decision-making. We demonstrate across three domains Risk game, CAD code generation, and arithmetic reasoning that our approach outperforms baselines, yielding higher accuracy and stronger structural alignment. Our contribution is to demonstrate that Const-of-T offers a generalizable foundation for constraint-guided reasoning, enabling more efficient, constraint-aligned, and domain-adaptable planning with LLMs.

Constrained decision-making problems are pervasive across various disciplines.

Continuous allocation tasks are a class of problems where an agent needs to distribute a limited amount of resources over a set of entities at each time step.