Combinatorial Optimization

Modern transportation network modeling increasingly involves the integration of diverse methodologies including sensor-based forecasting, reinforcement learning, classical flow optimization, and demand modeling that have traditionally been developed in isolation. This paper introduces Flow Through Tensors (FTT), a unified computational graph architecture that connects origin destination flows, path probabilities, and link travel times as interconnected tensors. Our framework makes three key contributions: first, it establishes a consistent mathematical structure that enables gradient-based optimization across previously separate modeling elements; second, it supports multidimensional analysis of traffic patterns over time, space, and user groups with precise quantification of system efficiency; third, it implements tensor decomposition techniques that maintain computational tractability for large scale applications. These innovations collectively enable real time control strategies, efficient coordination between multiple transportation modes and operators, and rigorous enforcement of physical network constraints. The FTT framework bridges the gap between theoretical transportation models and practical deployment needs, providing a foundation for next generation integrated mobility systems.

This paper presents a fully probabilistic approach for solving optimal experimental design problems under budget constraints. The experimental design is viewed as a random variable and is associated with a parametric conditional distribution that inherently models the budget constraints. The original optimization problem is replaced with an optimization over the expected value of the original objective, which is then optimized over the distribution parameters. The resulting optimal parameter (policy) is used to sample the feasible region of binary space to produce estimates of the optimal solution(s) of the original optimization problem. In this work we extend the family of conditional Bernoulli models to model the random variable conditioned by the total number of nonzero entries, that is, the budget constraint. This approach (a) is generally applicable to binary optimization problems with nonstochastic black-box objective functions and budget constraints; (b) employs conditional probabilities to model and sample only the feasible region and thus considerably reduces the computational cost compared with employing soft constraints; and (c) does not employ soft constraints and thus does not require tuning of a regularization parameter, for example to promote sparsity, which is generally challenging. The proposed approach is verified numerically using an optimal sensor placement experiment based on an advection-diffusion forward model in a parameter identification setup.

2D irregular packing is a classic combinatorial optimization problem with various applications, such as material utilization and texture atlas generation. Due to its NP-hard nature, conventional numerical approaches typically encounter slow convergence and high computational costs. Previous research (GFPack) introduced a generative method for gradient-based packing, providing early evidence of its feasibility but faced limitations such as insufficient rotation support, poor boundary adaptability, and high overlap ratios. In this paper, we propose GFPack++, a deeply investigated framework that adopts attention-based geometry and relation encoding, enabling more comprehensive modeling of complex packing relationships. We further design a constrained gradient and a weighting function to enhance both the feasibility of the produced solutions and the learning effectiveness. Experimental results on multiple datasets demonstrate that GFPack++ achieves higher space utilization, supports continuous rotation, generalizes well to arbitrary boundaries, and infers orders of magnitude faster than previous approaches. Codes for this paper are at https://github.com/TimHsue/GFPack-pp.

Deep neural networks based on reinforcement learning (RL) for solving combinatorial optimization (CO) problems are developing rapidly and have shown a tendency to approach or even outperform traditional solvers. However, existing methods overlook an important distinction: CO problems differ from other traditional problems in that they focus solely on the optimal solution provided by the model within a specific length of time, rather than considering the overall quality of all solutions generated by the model. In this paper, we propose Leader Reward and apply it during two different training phases of the Policy Optimization with Multiple Optima (POMO) model to enhance the model's ability to generate optimal solutions. This approach is applicable to a variety of CO problems, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), and the Flexible Flow Shop Problem (FFSP), but also works well with other POMO-based models or inference phase's strategies. We demonstrate that Leader Reward greatly improves the quality of the optimal solutions generated by the model. Specifically, we reduce the POMO's gap to the optimum by more than 100 times on TSP100 with almost no additional computational overhead.

This manuscript provides a comprehensive review of the Maximum Clique Problem, a computational problem that involves finding subsets of vertices in a graph that are all pairwise adjacent to each other. As such, this review is a continuation of the series of previous reviews from 1994, 1999 and 2014. The manuscript covers in a simple way classical algorithms and includes a review of recent developments in graph neural networks and quantum algorithms.

The design and organization of complex robotic systems traditionally requires laborious trial-and-error processes to ensure both hardware and software components are correctly connected with the resources necessary for computation. This paper presents a novel generalization of the quadratic assignment and routing problem, introducing formalisms for selecting components and interconnections to synthesize a complete system capable of providing some user-defined functionality. By introducing mission context, functional requirements, and modularity directly into the assignment problem, we derive a solution where components are automatically selected and then organized into an optimal hardware and software interconnection structure, all while respecting restrictions on component viability and required functionality. The ability to generate complete functional systems directly from individual components reduces manual design effort by allowing for a guided exploration of the design space. Additionally, our formulation increases resiliency by quantifying resource margins and enabling adaptation of system structure in response to changing environments, hardware or software failure. The proposed formulation is cast as an integer linear program which is provably $\mathcal{NP}$-hard. Two case studies are developed and analyzed to highlight the expressiveness and complexity of problems that can be addressed by this approach: the first explores the iterative development of a ground-based search-and-rescue robot in a variety of mission contexts, while the second explores the large-scale, complex design of a humanoid disaster robot for the DARPA Robotics Challenge. Numerical simulations quantify real world performance and demonstrate tractable time complexity for the scale of problems encountered in many modern robotic systems.

Combinatorial scheduling poses a significant challenge for language models, requiring them to identify feasible solutions within exponentially large search spaces while satisfying complex constraints. This challenge is especially pronounced in resource-constrained settings, where larger language models are impractical and selection is limited to smaller models which often fail to preserve feasibility when scheduling directly from natural language. To address these limitations, we introduce SDDL, a neuro-symbolic framework that translates natural-language scheduling problems into compact, solver-aligned representations of tasks, resources, constraints, and objectives, while delegating low-level modeling and search to a deterministic compiler and external solver. On a 300-instance, multi-family subset of scheduling problems, SDDL improves independently verified feasibility for every resource-constrained model tested. The two strongest SDDL configurations reach 55.3% and 28.3%, up from direct-generation baselines of 23.7% and 1.3% and solver-code baselines of 21.7% and 7.0%, with a 0.0% median optimality gap among feasible schedules. By expressing problem structure rather than generating solutions or solver code, SDDL enables smaller models to approach the strongest evaluated direct- and solver-code configurations, including substantially larger frontier models.