Safety verification ensures that a system avoids undesired behaviour. Liveness complements safety, ensuring that the system also achieves its desired objectives. A complete specification of functional correctness must combine both safety and liveness. Proving with mathematical certainty that a system satisfies a safety property demands presenting an appropriate inductive invariant of the system, whereas proving liveness requires showing a measure of progress witnessed by a ranking function. Neural model checking has recently introduced a data-driven approach to the formal verification of reactive systems, albeit focusing on ranking functions and thus addressing liveness properties only.

The Time-Dependent Traveling Salesman Problem (TDTSP) extends the classical TSP by allowing dynamic edge weights that vary with departure time, reflecting real-world scenarios such as transportation networks, where travel times fluctuate due to congestion patterns. TDTSP violates symmetry, triangle inequality, and cyclic invariance properties of classical TSP, creating unique computational challenges. In this paper, we propose a neural model that extends MatNet from static asymmetric TSP to time-dependent settings by using an adjacency tensor to capture temporal variations, followed by a time-aware decoder. Our architecture addresses the unique challenge of asymmetry and triangle inequality violations that change dynamically over time. Beyond architectural innovations, our research reveals a critical evaluation insight: many practical TDTSP instances maintain the same optimal solution regardless of time-dependent edge weights.

The constrained version of the standard online convex optimization (OCO) framework, called COCO is considered, where on every round, a convex cost function and a convex constraint function are revealed to the learner after it chooses the action for that round. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV). An algorithm is proposed that guarantees a static regret of O( T) and a CCV of min{V,O( Tlog T)}, where V depends on the distance between the consecutively revealed constraint sets, the shape of constraint sets, dimension of action space and the diameter of the action space. When constraint sets have additional structure, V = O(1). Compared to the state of the art results, static regret of O( T) and CCV of O( T log T), that were universal, the new result on CCV is instance dependent, which is derived by exploiting the geometric properties of the constraint sets.

While large language models (LLMs) have emerged as promising tools for CO--either by directly generating solutions or synthesizing solver-specific codes--existing approaches often neglect critical structural priors inherent to CO problems, leading to suboptimality and iterative inefficiency. Inspired by human experts' success in leveraging CO structures for algorithm design, we propose STRCMP, a novel structure-aware LLM-based algorithm discovery framework that systematically integrates structure priors to enhance solution quality and solving efficiency. Our framework combines a graph neural network (GNN) for extracting structural embeddings from CO instances with an LLM conditioned on these embeddings to identify high-performing algorithms in the form of solver-specific codes. This composite architecture ensures syntactic correctness, preserves problem topology, and aligns with natural language objectives, while an evolutionary refinement process iteratively optimizes generated algorithm. Extensive evaluations across Mixed Integer Linear Programming and Boolean Satisfiability problems, using nine benchmark datasets, demonstrate that our proposed STRCMPoutperforms five strong neural and LLM-based methods by a large margin, in terms of both solution optimality and computational efficiency.

The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. We propose a novel continuous relaxation framework for ATSP that leverages differentiable constraints to encourage acyclic structures and valid permutations.

We introduce BikeBench, an engineering design benchmark for evaluating generative models on problems with multiple real-world objectives and constraints. As generative AI's reach continues to grow, evaluating its capability to understand physical laws, human guidelines, and hard constraints grows increasingly important. Engineering product design lies at the intersection of these difficult tasks, providing new challenges for AI capabilities. BikeBench evaluates AI models' capabilities to generate bicycle designs that not only resemble the dataset, but meet specific performance objectives and constraints. To do so, BikeBench quantifies a variety of human-centered and multiphysics performance characteristics, such as aerodynamics, ergonomics, structural mechanics, human-rated usability, and similarity to subjective text or image prompts. Supporting the benchmark are several datasets of simulation results, a dataset of 10,000 human-rated bicycle assessments, and a synthetically generated dataset of 1.6M designs, each with a parametric, CAD/XML, SVG, and PNG representation. BikeBench is uniquely configured to evaluate tabular generative models, large language models (LLMs), design optimization, and hybrid algorithms side-by-side. Our experiments indicate that LLMs and tabular generative models fall short of hybrid GenAI+optimization algorithms in design quality, constraint satisfaction, and similarity scores, suggesting significant room for improvement. We hope that BikeBench, a first-of-its-kind benchmark, will help catalyze progress in generative AI for constrained multi-objective engineering design problems.

Discrete diffusion models are a class of generative models that construct sequences by progressively denoising samples from a categorical noise distribution. Beyond their rapidly growing ability to generate coherent natural language, these models present a new and important opportunity to enforce sequence-level constraints, a capability that current autoregressive models cannot natively provide. This paper capitalizes on this opportunity by introducing Constrained Discrete Diffusion (CDD), a novel integration of differentiable constraint optimization within the diffusion process to ensure adherence to constraints, logic rules, or safety requirements for generated sequences. Unlike conventional text generators that often rely on post-hoc filtering or model retraining for controllable generation, CDD directly imposes constraints into the discrete diffusion sampling process, resulting in a training-free and effective approach. Experiments in toxicity-controlled text generation, property-constrained molecule design, and instruction-constrained text completion demonstrate that CDD achieves zero constraint violations in a diverse array of tasks while preserving fluency, novelty, and coherence, while outperforming autoregressive and existing discrete diffusion approaches.

In constrained MDPs (CMDPs) with adversarial rewards and constraints, a known impossibility result prevents any algorithm from attaining sublinear regret and constraint violation, when competing against a best-in-hindsight policy that satisfies the constraints on average. In this paper, we show how to ease such a negative result, by considering settings that generalize both stochastic CMDPs and adversarial ones. We provide algorithms whose performances smoothly degrade as the level of environment adverseness increases. Specifically, they attain eO( T +C) regret and positive constraint violation under bandit feedback, where C measures the adverseness of rewards and constraints. This is C = Θ(T) in the worst case, coherently with the impossibility result for adversarial CMDPs. First, we design an algorithm with the desired guarantees when C is known. Then, in the case C is unknown, we obtain the same results by embedding multiple instances of such an algorithm in a general meta-procedure, which suitably selects them so as to balance the trade-off between regret and constraint violation.

Long-horizon planning in robotic manipulation requires translating under-specified, symbolic goals into executable control programs that satisfy spatial, temporal, and physical constraints.

Deep generative models have recently been applied to physical systems governed by partial differential equations (PDEs), offering scalable simulation and uncertainty-aware inference. However, enforcing physical constraints, such as conservation laws (linear and nonlinear) and physical consistencies, remains challenging. Existing methods often rely on soft penalties or architectural biases that fail to guarantee hard constraints. In this work, we propose Physics-Constrained Flow Matching (PCFM), a zero-shot inference framework that enforces arbitrary nonlinear constraints in pretrained flow-based generative models. PCFM continuously guides the sampling process through physics-based corrections applied to intermediate solution states, while remaining aligned with the learned flow and satisfying physical constraints. Empirically, PCFM outperforms both unconstrained and constrained baselines on a range of PDEs, including those with shocks, discontinuities, and sharp features, while ensuring exact constraint satisfaction at the final solution. Our method provides a flexible framework for enforcing hard constraints in both scientific and general-purpose generative models, especially in applications where constraint satisfaction is essential.