Graph Neural Networks (GNNs) have emerged as a promising approach for ``learning to branch'' in Mixed-Integer Linear Programming (MILP). While standard Message-Passing GNNs (MPNNs) are efficient, they theoretically lack the expressive power to fully represent MILP structures. Conversely, higher-order GNNs (like 2-FGNNs) are expressive but computationally prohibitive. In this work, we investigate Subgraph GNNs as a theoretical middle ground. Crucially, while previous work [Chen et al., 2025] demonstrated that GNNs with 3-WL expressive power can approximate Strong Branching, we prove a sharper result: node-anchored Subgraph GNNs whose expressive power is strictly lower than 3-WL [Zhang et al., 2023] are sufficient to approximate Strong Branching scores. However, our extensive empirical evaluation on four benchmark datasets reveals a stark contrast between theory and practice. While node-anchored Subgraph GNNs theoretically offer superior branching decisions, their $O(n)$ complexity overhead results in significant memory bottlenecks and slower solving times than MPNNs and heuristics. Our results indicate that for MILP branching, the computational cost of expressive GNNs currently outweighs their gains in decision quality, suggesting that future research must focus on efficiency-preserving expressivity.

We thank all the reviewers. We hope the reviewers could increase the rating if the response addressed your concerns. Theoretically, in Eq. (11), if we take Accordingly, we have revised the relevant sections of the paper by adding pertinent technical details. General Comments: All the learned models that we report performance about are certified monotonic. We will make a note of this in the paper; see also G#1 and G#2.

Following the recent push for trustworthy AI, there has been an increasing interest in developing contrastive explanation techniques for optimisation, especially concerning the solution of specific decision-making processes formalised as MILPs. Along these lines, we propose X-MILP, a domain-agnostic approach for building contrastive explanations for MILPs based on constraint reasoning techniques. First, we show how to encode the queries a user makes about the solution of an MILP problem as additional constraints. Then, we determine the reasons that constitute the answer to the user's query by computing the Irreducible Infeasible Subsystem (IIS) of the newly obtained set of constraints. Finally, we represent our explanation as a "graph of reasons" constructed from the IIS, which helps the user understand the structure among the reasons that answer their query. We test our method on instances of well-known optimisation problems to evaluate the empirical hardness of computing explanations.

We proposes a novel method that enables Graph Neural Networks (GNNs) to solve SAT problems by leveraging a technique developed for applying GNNs to Mixed Integer Linear Programming (MILP). Specifically, k-CNF formulae are mapped into MILP problems, which are then encoded as weighted bipartite graphs and subsequently fed into a GNN for training and testing. From a theoretical perspective: (i) we establish permutation and equivalence invariance results, demonstrating that the method produces outputs that are stable under reordering of clauses and variables; (ii) we identify a theoretical limitation, showing that for a class of formulae called foldable formulae, standard GNNs cannot always distinguish satisfiable from unsatisfiable instances; (iii) we prove a universal approximation theorem, establishing that with Random Node Initialization (RNI), the method can approximate SAT solving to arbitrary precision on finite datasets--that is, the GNN becomes approximately sound and complete on such datasets. Furthermore, we show that for unfoldable formulae, the same approximation guarantee can be achieved without the need for RNI. Finally, we conduct an experimental evaluation of our approach, which show that, despite the simplicity of the neural architecture, the method achieves promising results.

Composite indicators are widely used to score or classify units evaluated on multiple criteria. Their construction involves aggregating criteria evaluations, a common practice in Multiple Criteria Decision Aiding (MCDA). In MCDA, various methods have been proposed to address key aspects of multiple criteria evaluations, such as the measurement scales of the criteria, the degree of acceptable compensation between them, and their potential interactions. However, beyond producing a final score or classification, it is essential to ensure the explainability and interpretability of results as well as the procedure's transparency. This paper proposes a method for constructing explainable and interpretable composite indicators using " if..., then... " decision rules. We consider the explainability and interpretability of composite indicators in four scenarios: (i) decision rules explain numerical scores obtained from an aggregation of numerical codes corresponding to ordinal qualifiers; (ii) an obscure numerical composite indicator classifies units into quantiles; (iii) given preference information provided by a Decision Maker in the form of classifications of some reference units, a composite indicator is constructed using decision rules; (iv) the classification of a set of units results from the application of an MCDA method and is explained by decision rules. To induce the rules from scored or classified units, we apply the Dominance-based Rough Set Approach. The resulting decision rules relate the class assignment or unit's score to threshold conditions on values of selected indicators in an intelligible way, clarifying the underlying rationale. Moreover, they serve to recommend composite indicator assessment for new units of interest.

Mixed-Integer Linear Programming (MILP) is widely used in fields such as scheduling, logistics, and planning. Enhancing the performance of MILP solvers, particularly learning-based solvers, requires substantial amounts of high-quality data. However, existing methods for MILP instance generation typically necessitate training a separate model for each problem class and are computationally intensive when generating new instances. To address these limitations, we reformulate the MILP Instance Generation task as MILP Code Generation task, enabling efficient, flexible, and interpretable instance generation through code. Since MILP instances generated from code can vary significantly in scale, we introduce MILP-EmbedSim, a new similarity metric that accurately measures the similarity between instances of varying sizes within the same problem class. Leveraging this metric, we propose MILP-Retrieval, a pipeline that retrieves generation code from library to produce MILP instances highly similar to target instance. MILP-Retrieval outperforms baselines in both MILP Code Generation and Instance Generation tasks, provides a novel perspective on MILP instance generation and opens new possibilities for learning-based solvers.

This work presents novel algorithms for learning Bayesian networks of bounded treewidth. Both exact and approximate methods are developed. The exact method combines mixed integer linear programming formulations for structure learning and treewidth computation. The approximate method consists in sampling k-trees (maximal graphs of treewidth k), and subsequently selecting, exactly or approximately, the best structure whose moral graph is a subgraph of that k-tree. The approaches are empirically compared to each other and to state-of-the-art methods on a collection of public data sets with up to 100 variables.

By exploiting the correlation between the structure and the solution of Mixed-Integer Linear Programming (MILP), Machine Learning (ML) has become a promising method for solving large-scale MILP problems. Existing ML-based MILP solvers mainly focus on end-to-end solution learning, which suffers from the scalability issue due to the high dimensionality of the solution space. Instead of directly learning the optimal solution, this paper aims to learn a reduced and equivalent model of the original MILP as an intermediate step. The reduced model often corresponds to interpretable operations and is much simpler, enabling us to solve large-scale MILP problems much faster than existing commercial solvers. However, current approaches rely only on the optimal reduced model, overlooking the significant preference information of all reduced models. To address this issue, this paper proposes a preference-based model reduction learning method, which considers the relative performance (i.e., objective cost and constraint feasibility) of all reduced models on each MILP instance as preferences. We also introduce an attention mechanism to capture and represent preference information, which helps improve the performance of model reduction learning tasks. Moreover, we propose a SetCover based pruning method to control the number of reduced models (i.e., labels), thereby simplifying the learning process. Evaluation on real-world MILP problems shows that 1) compared to the state-of-the-art model reduction ML methods, our method obtains nearly 20% improvement on solution accuracy, and 2) compared to the commercial solver Gurobi, two to four orders of magnitude speedups are achieved.