Transformer-Based Flow Shop Scheduling Using MILP-Generated Training Data

Advances in machine learning (ML) have created new opportunities to complement traditional operations research (OR) methods. In particular, transformer models can capture complex interactions in token sequences by mapping tokens into a high-dimensional embedding space and propagating contextual information via attention. This makes them a candidate to model non-permutation flow shop scheduling with secondary resources as a next-token prediction task, where tokens represent job-machine-secondary resource tuples. For training, mixed-integer linear programming (MILP)-generated schedules are tokenized and used as next-token prediction data. During inference, partial token sequences (prefixes) are randomly generated and completed by the trained transformer through constrained decoding. A computational study is conducted on a flow shop with 8 jobs, 4 machines, and 3 secondary resources, where jobs are selected from a fixed pool of 20 jobs that is sampled during training and provides the candidates during prefix completion. The transformer achieves better solution quality (smaller makespans) compared to a genetic algorithm (GA), the NEH heuristic, and random search. It is outperformed only by the MILP model and the iterated greedy (IG) heuristic. The study concludes that transformer models can, to some extent, learn patterns from MILP-optimized non-permutation flow shop schedules and that transformer-based scheduling represents an interesting direction for future research, particularly in settings with a fixed, recurring job set.

SHSP: Structure-Aware Hierarchical Solution Prediction for Mixed-Integer Linear Programming

Mixed-Integer Linear Programming (MILP) is a fundamental optimization paradigm in combinatorial optimization and has been widely applied across real-world domains. Due to its NP-hard nature, obtaining optimal solutions for large-scale or highly constrained MILP instances remains computationally prohibitive. Learning-based solution prediction has therefore emerged as a promising approach to provide high-quality variable assignment for solver acceleration. However, existing methods typically adopt a one-shot prediction paradigm that predicts the marginal probabilities of all variables simultaneously. As a result, the conditional dependencies among variables are only implicitly captured through message passing, with the burden of modeling the combinatorial structure falling entirely on the representational capacity of graph neural networks. To address this limitation, we propose the Structure-Aware Hierarchical Solution Prediction (SHSP) framework that replaces the parallel marginal decoding of one-shot methods with a novel hierarchical conditional decoding mechanism. Specifically, SHSP constructs a variable coupling graph from the constraint structure, decodes variables sequentially along a hierarchy of increasing coupling strength, and conditions each hierarchy on previously predicted assignments. To mitigate error accumulation during the decoding process, SHSP further incorporates a confidence-aware mask-and-repair mechanism to identify and correct unreliable intermediate predictions. We integrate SHSP with multiple learning-guided search methods, and evaluate it on four standard MILP benchmarks. Experimental results demonstrate that SHSP significantly outperforms existing one-shot prediction baselines, achieving a 54% average reduction in solution gap.

Learning Early-to-Final Solution Consistency for MILP Acceleration

Mixed-Integer Linear Programming (MILP) is a fundamental problem class in operations research and combinatorial optimization, with broad applications to industrial decision-making. Owing to their NP-hardness, however, modern solvers may struggle to find high-quality solutions for challenging MILP instances within practical time limits. Recent learning-based approaches seek to accelerate MILP solving by directly predicting high-quality solutions from static instance-level features, such as variable-constraint bipartite graphs. Yet accurate solution prediction from instance features alone is difficult, and these methods largely overlook the information revealed during the solver's search process. In this paper, we find that solutions produced at the early search stage of MILP solvers, which are computationally cheap to obtain, are often structurally close to the solutions found after full-budget search. Motivated by this observation, we propose a new solver-informed paradigm that shifts the learning target from variable assignment to early-to-final consistency: for each variable, we predict whether its early-stage assignment should persist in full-budget solutions. The predicted consistency naturally guides downstream search, for instance by fixing the assignments deemed consistent. At inference time, we further ensemble consistency predictions across multiple early-stage solutions to improve robustness. Experiments across four MILP benchmarks show our method improves prediction-guided search across diverse downstream pipelines. With Gurobi, our proposed method reduces the primal gap by 56.9% on average and closes it completely on combinatorial auction instances. Besides, we transferred the Gurobi-trained model zero-shot to SCIP without adaptation, achieving a 36.4% average gap reduction across benchmarks.

A Weisfeiler-Leman Characterization of Global-Attention Graph Transformers for Mixed-Integer Linear Programs

Graph foundation models (GFMs) with global attention are increasingly used to represent mixed-integer linear programs (MILPs), aiming to capture structure beyond the locality of standard graph neural networks. We study their expressive power through graph isomorphism testing, asking which MILP instances they map to identical representations. We prove that a broad class of hierarchical graph transformers combining global linear attention, edge-weighted cross-attention, and bipartite message passing is bounded by the one-dimensional Weisfeiler-Leman (1-WL) test: under any parameter setting, 1-WL-equivalent MILP graphs receive identical graph embeddings. Our compositional proof shows that each architectural component is a symmetric multiset function and thus preserves 1-WL equivalence. We validate this characterization across ten diverse graph encoders, including Graphormer-, GraphGPS-, Set-Transformer-, and Gasse-style models. Across model capacities, graph scales, and pooling operators, every tested encoder maps 1-WL-equivalent non-isomorphic graph pairs to numerically identical embeddings. Consequently, graph invariants that vary within a 1-WL equivalence class cannot be recovered from these representations. We further show that expressiveness beyond 1-WL arises from input encoding rather than attention: random-walk positional encodings separate the constructed pairs, while additional constructions expose the limits of this remedy. These results characterize the expressive power of global-attention GFMs and provide an encoder-agnostic diagnostic for detecting 1-WL-induced representation equivalence.

Low-Latency Relay Selection in NR-V2X Vehicular Communications via Graph Isomorphism Networks with Edge Features

Reliable, low-latency uplink connectivity is a key requirement for C-V2X networks in dense urban environments, where fast channel variations and blockages often degrade direct vehicle-to-infrastructure links. Multi-hop relaying can restore coverage, but relay-link activation under radio, capacity, and routing constraints results in an NP-hard optimisation problem, typically solved via Mixed-Integer Linear Programming (MILP), whose runtime scales poorly with graph size. This paper introduces an edge-aware Learning-to-Optimise framework for real-time relay selection. Each V2X snapshot is modelled as a directed graph: node features encode vehicle state and traffic demand, while edge features capture radio-link capacity. An offline MILP oracle generates optimal relay configurations that supervise a Graph Isomorphism Network with Edge Features (GINE), enabling edge-level relay activation through a single forward pass, with tightly bounded inference latency. To bridge learning and exact optimisation, we also propose a hybrid GINE-Pruned MILP (GP-MILP) strategy in which GINE predictions prune the MILP search space. Experiments on a large-scale dataset generated via an OSM-SUMO-GEMV$^2$ pipeline show that GINE closely matches MILP decisions at the link level (accuracy 0.9589), F1-score (0.9544) on validation) and yields consistent end-to-end connectivity gains over a 1-hop MILP baseline (up to 9.2% with four RSUs and 12% with two RSUs). Inference latency remains tightly bounded, with all evaluated instances completing within 5~ms. Moreover, GP-MILP preserves MILP-equivalent solutions (same objective value) while achieving solver runtimes below 30~ms for more than 98%) of the graph instances, making MILP-grade optimisation compatible with stringent NR-V2X latency budgets.

LP Mining with LP2Graph: A Use Case for Railway Rescheduling

Like many optimization-driven domains, railway rescheduling relies on Mixed-Integer Linear Programming (MILP), yet the field's modeling knowledge is scattered across hundreds of papers in incompatible notations, and narrative surveys organize it subjectively: they classify models by vocabulary rather than by structure, and reproduce neither. We present LP Mining with LP2Graph, a method that mines the structure of published LP and MILP formulations into a reproducible dataset and an induced taxonomy. Its core, LP2Graph, represents each formulation admitted by its canonical grammar as a typed variable--equation graph derived from a single canonical model; once a source is extracted into that model, everything downstream is deterministic. Each source is parsed into this model, homologized, and clustered bottom-up (over variables, then constraints and the objective, then whole-model structure) and, separately, by application domain and solution approach; the resulting groups are labeled by a rule-seeded, self-updating classifier. We validate the representation rather than assume it: per-cluster representatives are regenerated as independent LaTeX and re-solved across CBC, HiGHS and Gurobi against the optimum reported in the source paper. The outcome is an objective, repeatable taxonomy of variables, constraints and model types: the principled foundation on which our raiLPminer line of automated railway-rescheduling model development builds.

Sim2Schedule: A Simulator-Guided LLM Framework for Autonomous Open-Pit Mine Scheduling

Open-pit mine scheduling is a critical process for maximizing economic return under complex geotechnical and operational constraints. While Mixed-Integer Linear Programming (MILP) provides mathematically optimal baselines, its exponential computational complexity and inability to adapt in real time limit its practical deployment in dynamic industrial environments. This work introduces a simulator-driven Large Language Model (LLM) scheduling framework in which the LLM acts as an autonomous decision-making agent, guided at each step by a custom simulator that encodes geotechnical precedence, extraction-processing coupling, and dynamic capacity constraints directly into the action generation mechanism. Operating entirely zero-shot within a closed, data-secure environment, the framework produces complete, interpretable extraction and processing schedules without cloud-based inference, domain-specific fine-tuning, or retraining. To provide a trustworthy performance benchmark, a novel MILP formulation is developed that incorporates realistic operational and geotechnical constraints. Evaluated across mining instances of varying scale and time periods, the LLM-based framework recovers between 94\% and 99\% of the MILP optimal NPV while scaling linearly in computation time. These results position simulator-constrained LLM agents as a practical and scalable alternative to classical optimization for long-horizon industrial scheduling under complex operational constraints.