Combinatorial Optimization

We propose a learning-augmented framework for accelerating max-flow computation and image segmentation by integrating Graph Neural Networks (GNNs) with the Ford-Fulkerson algorithm. Rather than predicting initial flows, our method learns edge importance probabilities to guide augmenting path selection. We introduce a Message Passing GNN (MPGNN) that jointly learns node and edge embeddings through coupled updates, capturing both global structure and local flow dynamics such as residual capacity and bottlenecks. Given an input image, we propose a method to construct a grid-based flow network with source and sink nodes, extract features, and perform a single GNN inference to assign edge probabilities reflecting their likelihood of belonging to high-capacity cuts. These probabilities are stored in a priority queue and used to guide a modified Ford-Fulkerson procedure, prioritizing augmenting paths via an Edmonds-Karp-style search with bottleneck-aware tie-breaking. This avoids repeated inference over residual graphs while leveraging learned structure throughout optimization. We further introduce a bidirectional path construction strategy centered on high-probability edges and provide a theoretical framework relating prediction quality to efficiency via a weighted permutation distance metric. Our method preserves max-flow/min-cut optimality while reducing the number of augmentations in practice. We also outline a hybrid extension combining flow warm-starting with edge-priority prediction, establishing a foundation for learning-guided combinatorial optimization in image segmentation.

The recent emergence of planar transport systems necessitates re-evaluation of Flexible Manufacturing Systems (FMS) to address the simultaneous scheduling of internal logistics and production operations. By operating on a tile-based planar grid, these systems allow independent movers full two-dimensional freedom, mitigating inefficiencies inherent to traditional sequential lines. This paper applies a planar FMS framework to a real-world use case in the pharmaceutical industry: the automated production of personalized drugs. Implementing this system requires solving optimization problems at both tactical and operational levels. The tactical level involves decisions regarding production line layout and the positioning of drug dispensers. A Mixed-Integer Quadratic Programming model is utilized for the packing problem to exploit drug co-occurrence patterns found in historical patient data. Subsequently, we solve the placement problem - a bi-level problem combining an assignment problem with Shortest Hamiltonian paths with neighborhoods - to arrange dispensers in a layout minimizing expected travel distances. The operational level is encountered daily, scheduling individual movers to process new orders as quickly as possible. This scheduling problem is formulated using Constraint Programming, modeling movers as reservoir resources to ensure order completeness, complemented by a routing phase using an iterative conflict-resolution mechanism and DAG-based reasoning to convert schedules into conflict-free paths. Evaluation using real-world prescription data for 40 drugs shows the framework scales efficiently across several layout topologies for up to 500 orders, with schedules that are highly effective and computationally tractable for daily operations.

High-performance TSP solvers such as Lin-Kernighan-Helsgaun (LKH) search within a \emph{candidate graph} -- a small subset of edges pre-selected for the solver -- rather than over the complete graph. The two leading sparsification heuristics, $α$-Nearest and POPMUSIC, each fall short of the density-coverage balance: $α$-Nearest is dense with stable recall, while POPMUSIC is sparser but its recall degrades with scale. Their union closes the recall gap while remaining far below the complete graph in density, leaving room for further reduction. Existing learning-based sparsifiers score edges on the complete graph, an approach that is expensive and largely limited to Euclidean instances. We propose a two-stage method that inverts this logic. Stage1 takes the union of $α$-Nearest and POPMUSIC, achieving near-perfect recall at ${\sim}6N$ edges. Crucially, the union annotates each edge with its \emph{source provenance} -- whether it was endorsed by $α$-Nearest, POPMUSIC, or both. Stage2 trains a lightweight classifier on these annotated edges and prunes the lowest-scoring ones. Because dual-source edges are almost always optimal, the learning problem reduces to filtering the single-source subset -- a substantially easier task than classifying all $O(N^2)$ edges from scratch. Across four distance types, five spatial distributions, and problem sizes from 50 to 500, the pipeline reduces candidate-graph density by $37$-$47\%$ while retaining ${\geq}99.69\%$ of optimal-tour edges, and matches or exceeds the coverage of recent Euclidean-only neural sparsifiers at lower density at TSP500.

The quadratic assignment problem (QAP) is a fundamental NP-hard task that poses significant challenges for both traditional heuristics and modern learning-based solvers. Existing QAP solvers still struggle to achieve consistently competitive performance across structurally diverse real-world instances. To bridge this performance gap, we propose PLMA, an innovative permutation learning framework. PLMA features an efficient warm-started MCMC finetuning procedure to enhance deployment-time performance, leveraging short Markov chains to anchor the adaptation to the promising regions previously explored. For rapid exploration via MCMC over the permutation space, we design an additive energy-based model (EBM) that enables an $O(1)$-time 2-swap Metropolis-Hastings sampling step. Moreover, the neural network used to parameterize the EBM incorporates a scalable and flexible cross-graph attention mechanism to model interactions between facilities and locations in the QAP. Extensive experiments demonstrate that PLMA consistently outperforms state-of-the-art baselines across various benchmarks. In particular, PLMA achieves a near-zero average optimality gap on QAPLIB, exhibits remarkably superior robustness on the notoriously difficult Taixxeyy instances, and also serves as an effective QAP solver in bandwidth minimization.

Scalability of evolutionary algorithms refers to assessing how their performance changes as problem size increases. In the area of multi-objective optimisation, research on the scalability of multi-objective evolutionary algorithms (MOEAs) has predominantly focussed on continuous problems. However, multi-objective combinatorial optimisation problems (MOCOPs) differ from continuous ones. Their discrete and rigid structure often brings rugged landscape, numerous local optimal solutions and disjoint global optimal regions. This leads to different behaviour of MOEAs. For example, SEMO, a simple MOEA without mating selection and diversity maintenance mechanisms, has been shown to be highly competitive, and in many cases to outperform more sophisticated MOEAs on MOCOPs. Yet, it remains unclear whether such findings hold for large-scale cases. In this paper, we conduct an empirical investigation into the scalability of MOEAs on combinatorial problems, with problem size from 50 to 5,000. Our results show that SEMO experiences a decline in convergence speed as dimensionality increases, compared to other MOEAs such as NSGA-II, SMS-EMOA and MOEA/D. We further demonstrate that the absence of crossover is a major contributor to SEMO's underperformance in large-scale problems, and that incorporating crossover into SEMO can substantially accelerate convergence in general, despite being detrimental in spreading solutions over the Pareto front.

A key challenge in distributed coalition formation within characteristic function games is determining how to allocate the calculation of coalition values across a set of agents. The number of possible coalitions grows exponentially with the number of agents, and existing distributed approaches may produce uneven or redundant allocations, or assign coalitions to agents that are not themselves members. In this article, we present the \emph{Necklace-based Distributed Coalition Algorithm} (N-DCA), a communication-free algorithm in which each agent independently determines its own coalition value calculation allocation using only its identifier and the total number of agents. The approach builds on the notion of Increment Arrays (IAs), for which we develop a complete mathematical framework: equivalence classes under circular shifts, periodic IAs, and a rotated designation scheme with formal load-balance guarantees (tight bounds). We establish a bijection between canonical representative IAs and two-colour combinatorial necklaces, enabling the use of efficient necklace generation algorithms to enumerate allocations in constant amortised time. N-DCA is, to the best of our knowledge, the only distributed coalition value calculation algorithm for unrestricted characteristic function games to provably satisfy five desirable properties: no inter-agent communication, equitable allocation, no redundancy, balanced load, and self-interest. An empirical evaluation against DCVC (Rahwan and Jennings 2007) demonstrates that, although DCVC is faster by a constant factor, this difference becomes negligible under realistic characteristic-function evaluation costs, while N-DCA offers advantages in working memory, scalability, and the self-interest guarantee.

Combinatorial black-box optimization in high-dimensional settings demands a careful trade-off between exploiting promising regions of the search space and preserving sufficient exploration to identify multiple optima. Although Estimation-of-Distribution Algorithms (EDAs) provide a powerful model-based framework, they often concentrate on a single region of interest, which may result in premature convergence when facing complex or multimodal objective landscapes. In this work, we incorporate the Stein operator to introduce a repulsive mechanism among particles in the parameter space, thereby encouraging the population to disperse and jointly explore several modes of the fitness landscape. Empirical evaluations across diverse benchmark problems show that the proposed method achieves performance competitive with, and in several cases superior to, leading state-of-the-art approaches, particularly on large-scale instances. These findings highlight the potential of Stein variational gradient descent as a promising direction for addressing large, computationally expensive, discrete black-box optimization problems.

In many real-world settings, problem instances that need to be solved are quite similar, and knowledge from previous optimization runs can potentially be utilized. We explore this for the Traveling Salesperson problem with time windows (TSPTW), which often arises in settings where the travel-time matrix is fixed but time-window constraints change across related tasks. Existing TSPTW studies, however, have not systematically compared solving such task sequences independently with sequential transfer from previously solved tasks. We address this gap using a multi-task benchmark in which each base instance is expanded into five related tasks under two environments: partial time-window expansion and swap-additive time reassignment. We compare a standard from-scratch protocol with an iterative protocol that initializes each task from the best tour of the previous task, using the popular local search approaches LNS, VNS, and LKH-3 under a common penalized-score objective. Our experimental results show that the iterative protocol is consistently superior in the progressive-relaxation setting and generally competitive under swap-additive changes, with improvements increasing on more difficult instances.

Deep reinforcement learning (DRL) has recently emerged as a promising tool for Dynamic Algorithm Configuration (DAC), enabling evolutionary algorithms to adapt their parameters online rather than relying on static tuned configurations. While DRL can learn effective control policies, training is computationally expensive. This cost may be justified if learned policies generalize, allowing the training effort to transfer across instance types and problem scales. Yet, for real-world optimization problems, it remains unclear whether this promise holds in practice and under which conditions the investment in learning pays off. In this work, we investigate this question in the context of the carbon-aware permutation flow-shop scheduling problem. We develop a DRL-based DAC framework and train it exclusively on small, simple instances. We then deploy the learned policy on both similar and more complex unseen instances and compare its performance against a static tuned baseline, which provides a fair point of comparison. Our findings show that the proposed method provides a strong dynamic algorithm control policy that can be effectively transferred to different unseen problem instances. Notably, on simple and cheap to compute instances, similar to those observed during training and tuning, DRL performs comparably with the statically tuned baseline. However, as instance characteristics diverge and computational complexities increase, the DRL-learned policy continuously outperforms static tuning. These results confirm that DRL can acquire robust and generalizable control policies which are effective beyond the training instance distributions. This ability to generalize across instance types makes the initial computational investment worthwhile, particularly in settings where static tuning struggles to adapt to changing problem scenarios.

The use of satellite networks has increased significantly in recent years due to their advantages over purely terrestrial systems, such as higher availability and coverage. However, to effectively provide these services, satellite networks must cope with the continuous orbital movement and maneuvering of their nodes and the impact on the network's topology. In this work, we address the problem of (dynamic) network topology configuration under the online learning framework. As a byproduct, our approach does not assume structure about the network, such as known orbital planes (that could be violated by maneuvering satellites). We empirically demonstrate that our problem formulation matches the performance of state-of-the-art offline methods. Importantly, we demonstrate that our approach is amenable to constrained online learning, exhibiting a trade-off between computational complexity per iteration and convergence to a final strategy.

Autonomous AI agents are becoming economic actors. They compose deadline-bound services across the device-edge-cloud continuum and contend for capacity no single operator owns. This article asks when decentralised pricing can coordinate them exactly and truthfully, and what their service dependencies must satisfy. We model the pipelines as service-dependency graphs under governance constraints and identify the decisive property: laminarity of the induced leaf-block family. Where it holds, the feasible allocation space is a polymatroid and efficient incentive-compatible clearing exists per epoch; where a crossing breaks it, exactness and truthfulness can fail, while clearing prices survive on every totally unimodular family. The formal results instantiate polymatroid, gross-substitutes and Vickrey-Clarke-Groves theory; the contribution is the bridge from dependency structure to that machinery and an integrator architecture that exposes a laminar interface. The node-level evaluation (32,020 runs), at testbed-calibrated congestion, locates the boundary. One crossing costs exactness in 14.7% of high-load rounds and none on laminar instances, and truthfulness fails where exactness does, across contention. An integrator's inner exposure restores exactness at a cost in admitted volume, and before translation overhead it cuts median latency by 16 to 17 percent. Against truth-free posted prices the tuned market leads at medium load and trails a demand-responsive one in four of six high-load cells. Recorded and composed agent workloads drive the allocation. For platform designers, laminar pipelines admit exact decentralised pricing, truthful under the VCG mechanism, and a crossing needs an integrator's inner exposure.

Clustering is a central primitive in unsupervised learning, yet practice is dominated by heuristics whose outputs can be unstable and highly sensitive to representations, hyperparameters, and initialisation. Existing theoretical results are largely objective-specific and do not explain these behaviours at a unifying level. We formalise the common optimisation core underlying diverse clustering paradigms by defining the Universal Clustering Problem (UCP): the maximisation of a polynomial-time computable partition utility over a finite metric space. We prove the NP-hardness of UCP via two independent polynomial-time reductions from graph colouring and from exact cover by 3-sets (X3C). By mapping ten major paradigms -- including k-means, GMMs, DBSCAN, spectral clustering, and affinity propagation -- to the UCP framework, we demonstrate that each inherits this fundamental intractability. Our results provide a unified explanation for characteristic failure modes, such as local optima in alternating methods and greedy merge-order traps in hierarchical clustering. Finally, we show that clustering limitations reflect interacting computational and epistemic constraints, motivating a shift toward stability-aware objectives and interaction-driven formulations with explicit guarantees.

This work considers a number of optimization problems and reductive relations between them. The two main problems we are interested in are the Optimal Decision Tree and Set Cover. We study these two fundamental tasks under precedence constraints, that is, if a test (or set) $X$ is a predecessor of $Y$, then in any feasible decision tree $X$ needs to be an ancestor of $Y$ (or respectively, if $Y$ is added to set cover, then so must be $X$). For the Optimal Decision Tree we consider two optimization criteria: worst case identification time (height of the tree) or the average identification time. Similarly, for the Set Cover we study two cost measures: the size of the cover or the average cover time. Our approach is to develop a number of algorithmic reductions, where an approximation algorithm for one problem provides an approximation for another via a black-box usage of a procedure for the former. En route we introduce other optimization problems either to complete the `reduction landscape' or because they hold the essence of combinatorial structure of our problems. The latter is brought by a problem of finding a Maximum Density Precedence-Closed Subfamily, where the density is defined as the ratio of the number of items the family covers to its size. We provide $\mathcal{O}^*(\sqrt{m})$-approximation polynomial-time algorithms for all aforementioned problems. The picture is complemented by a number of hardness reductions that provide $\mathcal{O}(m^{1/12-ε})$-inapproximability results for the decision tree and covering problems. Besides giving a complete set of results for general precedence constraints, we also provide polylogarithmic approximation guarantees for two most typically studied and applicable graph types, outforests and inforests. By providing corresponding hardness results, we show most of these results to be tight.

The Winner Determination Problem (WDP) in combinatorial auctions is NP-hard, and no existing method reliably predicts which instances will defeat fast greedy heuristics. The ML-for-combinatorial-optimization community has focused on learning to \emph{replace} solvers, yet recent evidence shows that graph neural networks (GNNs) rarely outperform well-tuned classical methods on standard benchmarks. We pursue a different objective: learning to predict \emph{when} a given instance is hard for greedy allocation, enabling instance-dependent algorithm selection. We design a 20-dimensional structural feature vector and train a lightweight MLP hardness classifier that predicts the greedy optimality gap with mean absolute error 0.033, Pearson correlation 0.937, and binary classification accuracy 94.7% across three random seeds. For instances identified as hard -- those exhibiting ``whale-fish'' trap structure where greedy provably fails -- we deploy a heterogeneous GNN specialist that achieves ${\approx}0\%$ optimality gap on all six adversarial configurations tested (vs.\ 3.75--59.24% for greedy). A hybrid allocator combining the hardness classifier with GNN and greedy solvers achieves 0.51% overall gap on mixed distributions. Our honest evaluation on CATS benchmarks confirms that GNNs do not outperform Gurobi (0.45--0.71 vs.\ 0.20 gap), motivating the algorithm selection framing. Learning \emph{when} to deploy expensive solvers is more tractable than learning to replace them.

Neural networks, particularly message-passing neural networks (MPNNs), are increasingly used as heuristics for hard combinatorial optimization problems. Yet many learning-based methods rely on supervision, reinforcement learning, or gradient estimators, causing high computational cost, unstable training, or limited guarantees. Classical approximation algorithms provide worst-case guarantees but are non-differentiable and cannot adapt to structure in natural input distributions. We study this tradeoff through Uniform Facility Location (UniFL), a problem with applications in clustering, summarization, logistics, and supply chains. We propose a fully differentiable MPNN that incorporates approximation-algorithmic principles without solver supervision or discrete relaxations. The model has provable approximation guarantees and empirically improves on standard approximation algorithms, narrowing the gap to integer linear programming.

LLM-guided evolutionary computation, most notably AlphaEvolve, has been remarkably successful in discovering novel mathematical constructions by solving challenging optimization problems. The standard approach is to evolve a monolithic program that directly outputs a candidate solution. We present ImprovEvolve, an algorithmic alternative that drastically reduces cognitive load on the LLM. Instead of prompting the model for an end-to-end optimizer, we evolve a program with three specialized operators of initialization, local improvement, and perturbation. We then approach the optimum by iteratively applying local improvements and intensity-scheduled perturbations, effectively driving a basin-hopping search with LLM-evolved subroutines. For hexagon in hexagon packing, ImprovEvolve discovers new state-of-the-art packings of 11, 12, 15, and 16 hexagons, and additionally for 14, 17, and 23 hexagons after minimal expert tuning of the generated code. For the second autocorrelation inequality, the evolved and human-scaled program pushes the lower bound from 0.96102 to 0.96258. For spherical codes, the ImprovEvolve program lowers the best-known maximum cosine for the majority of 90 randomly chosen diverse state-of-the-art spherical codes, achieving relative improvements of up to 2.4%.

We study a stylized model of infrastructure investment in public transportation. In our model, each agent travels between a pair of terminals in a network captured by a weighted graph, where edge weights represent distances. The central planner can reduce the travel time along a fixed number of edges, with the goal of maximizing the utilitarian or egalitarian welfare. When there is only one agent, we provide a polynomial-time algorithm that combines Dijkstra's algorithm with a dynamic program. We then demonstrate how to use this algorithm as a subroutine to solve the problem for two agents. Generalizing this idea, we present an XP algorithm parameterized by the number of agents $n$; however, our problem turns out to be W[1]-hard with respect to $n$. Nevertheless, we establish a fixed-parameter tractability result for the special case where all agents travel to a common hub. If the number of agents is variable, we obtain NP-completeness and inapproximability results. We discuss implications of our results for a related model of railway network design.

The Gilbert-Pollak Conjecture \citep{gilbert1968steiner}, also known as the Steiner Ratio Conjecture, states that for any finite point set in the Euclidean plane, the Steiner minimum tree has length at least $\sqrt{3}/2 \approx 0.866$ times that of the Euclidean minimum spanning tree (the Steiner ratio). A sequence of improvements through the 1980s culminated in a lower bound of $0.824$, with no substantial progress reported over the past three decades. Recent advances in LLMs have demonstrated strong performance on contest-level mathematical problems, yet their potential for addressing open, research-level questions remains largely unexplored. In this work, we present a novel AI system for obtaining tighter lower bounds on the Steiner ratio. Rather than directly prompting LLMs to solve the conjecture, we task them with generating rule-constrained geometric lemmas implemented as executable code. These lemmas are then used to construct a collection of specialized functions, which we call verification functions, that yield theoretically certified lower bounds of the Steiner ratio. Through progressive lemma refinement driven by reflection, the system establishes a new certified lower bound of 0.8559 for the Steiner ratio. The entire research effort involves only thousands of LLM calls, demonstrating the strong potential of LLM-based systems for advanced mathematical research.

Graph neural networks are usually treated as auxiliaries for combinatorial optimization: they imitate algorithms, guide search, or supply scores to classical procedures. We show that this auxiliary role is not intrinsic. A GNN can itself be a heuristic. For the Euclidean Travelling Salesman Problem, we train a non-autoregressive GNN with no labels, rewards, sequential decoding, search, or local improvement. A differentiable Hamiltonian-cycle objective is the only supervision. The trained model produces a complete tour in one forward pass, while dropout and snapshots from a single training trajectory provide solution diversity without engineered moves. The heuristic is therefore learned, not programmed. It is also fast: batched inference remains in the millisecond regime on GPUs. Experiments on TSP100, TSP200, and TSP500 show that the model consistently improves over nearest-neighbor greedy baselines. These results identify unsupervised GNNs as a class of fast learned heuristics for combinatorial optimization.

Neural Combinatorial Optimization (NCO) has mostly focused on learning policies, typically neural networks, that operate on a single candidate solution at a time, either by constructing one from scratch or iteratively improving it. In contrast, decades of work in metaheuristics have shown that maintaining and evolving populations of solutions improves robustness and exploration, and often leads to stronger performance. To close this gap, we study how to make NCO explicitly population-based by learning policies that act on sets of candidate solutions. We first propose a simple taxonomy of population awareness levels and use it to highlight two key design challenges: (i) how to represent a whole population inside a neural network, and (ii) how to learn population dynamics that balance intensification (generating good solutions) and diversification (maintaining variety). We make these ideas concrete with two complementary tools: one that improves existing solutions using information shared across the whole population, and the other generates new candidate solutions that explicitly balance being high-quality with diversity. Experimental results on Maximum Cut and Maximum Independent Set indicate that incorporating population structure is advantageous for learned optimization methods and opens new connections between NCO and classical population-based search.

In this paper, we investigate the Resource-Constrained Project Scheduling Problem (RCPSP) with Time-of-Use (TOU) energy tariffs and machine states, a variant of RCPSP for production scheduling, where energy price is part of the criteria and one highly energy-demanding machine can be in one of the following three states: proc, idle, or off. The problem involves scheduling all tasks, respecting precedence constraints and resource limitations, while minimizing the combination of the overall makespan and the Total Energy Cost (TEC), which varies according to the TOU tariffs, which can take negative values. We propose two novel approaches to solve it: a monolithic Constraint Programming (CP) approach and a Logic-Based Benders Decomposition (LBBD) approach. The latter combines a master problem handling the energy cost solved using Integer Linear Programming (ILP) with a subproblem handling the RCPSP, resolved using CP. Both approaches outperform the monolithic compact ILP counterpart, but the LBBD significantly outperforms the monolithic CP in most cases, especially when the makespan criterion is not included in the objective function, solving to optimality instances with up to 480 tasks. Finally, we propose a way to generalize our LBBD approach to other problems sharing similar characteristics, and applied it to various problems, such as an RCPSP with blocking times & total weighted tardiness criterion, or a flexible job shop.

Anticlustering is an NP-hard combinatorial optimization problem that consists of partitioning a set of objects into equal-sized groups called anticlusters such that the objects in the same anticluster are as dissimilar as possible and thereby representative of the entire set of objects. Here we study the case where the dissimilarity metric is the squared Euclidean distance between the respective feature vectors. Applications of Euclidean anticlustering include social studies, cross-validation, creating mini-batches for stochastic gradient descent, and finding balanced K-cut partitions. In particular, machine-learning applications such as mini-batch generation involve million-scale datasets and very large values of K, making scalable anticlustering algorithms essential. We propose a new algorithm, the Assignment-Based Anticlustering (ABA) algorithm, that scales to instances with millions of objects and hundreds of thousands of anticlusters within seconds to minutes, which is far beyond what existing anticlustering methods can manage. We demonstrate here, via an extensive computational study, that our algorithm outperforms existing anticlustering methods in both solution quality and running time. This is so also for anticlustering with categories. For the related problem of balanced K-cut partitioning, our algorithm is superior to the well-known METIS method. The code of our algorithm is available on GitHub.

Realizing the full potential of quantum computation requires Quantum Error Correction (QEC). QEC reduces error rates by encoding logical information across redundant physical qubits, enabling errors to be detected and corrected. A common decoder used for this task is Minimum Weight Perfect Matching (MWPM) a graph-based algorithm that relies on edge weights to identify the most likely error chains. In this work, we propose a data-driven decoder named Neural Minimum Weight Perfect Matching (NMWPM). Our decoder utilizes a hybrid architecture that integrates Graph Neural Networks (GNNs) to extract local syndrome features and Transformers to capture long-range global dependencies, which are then used to predict dynamic edge weights for the MWPM decoder. To facilitate training through the non-differentiable MWPM algorithm, we formulate a novel proxy loss function that enables end-to-end optimization. Our findings on the toric code under depolarizing noise demonstrate thresholds of 17.9% and 10.95%, nearing the 18.9% and 11.0% maximum likelihood bounds, highlighting the advantage of hybrid decoders that combine the predictive capabilities of neural networks with the algorithmic structure of classical matching.

In the era of AI-driven science and engineering, we often want to design discrete objects in silico according to user-specified properties. For example, we may wish to design a protein to bind its target, arrange components within a circuit to minimize latency, or find materials with certain properties. Given a property predictive model, in silico design typically involves training a generative model over the design space (e.g., protein sequence space) to concentrate on designs with the desired properties. Distributional optimization$\unicode{x2013}$which can be formalized as an estimation of distribution algorithm or as reinforcement learning policy optimization$\unicode{x2013}$finds the generative model that maximizes an objective function in expectation. Optimizing a distribution over discrete-valued designs is in general challenging because of the combinatorial nature of the design space. However, many property predictors in scientific applications are decomposable in the sense that they can be factorized over design variables in a way that could in principle enable more effective optimization. For example, amino acids at a catalytic site of a protein may only loosely interact with amino acids of the rest of the protein to achieve maximal catalytic activity. Current distributional optimization algorithms are unable to make use of such decomposability structure. Herein, we propose and demonstrate use of a new distributional optimization algorithm, Decomposition-Aware Distributional Optimization (DADO), that can leverage any decomposability defined by a junction tree on the design variables, to make optimization more efficient. At its core, DADO employs a soft-factorized "search distribution"$\unicode{x2013}$a learned generative model$\unicode{x2013}$for efficient navigation of the search space, invoking graph message-passing to coordinate optimization across linked factors.

Column generation and branch-and-price (B&P) are leading mathematical optimization methods for large-scale exact optimization, iterating between solving a master problem and a pricing problem. Due to the difficulty of discrete optimization, high-performance column generation often relies on a custom pricing algorithm built specifically to exploit the problem's structure. This bespoke nature of the pricing solver makes column generation a problem-specific method and hinders the use of generic implementations across a wide range of problems. We show that domain-independent dynamic programming (DIDP), a model-based paradigm for dynamic programming, can be used as a generic pricing solver. We develop new modeling features and a solving algorithm for DIDP to achieve better performance in typical pricing problems. We demonstrate that in four problem classes, our implementations of B&P, with pricing by DIDP, empirically outperform an existing automated B&P solver and B&P with pricing by mixed-integer programming or constraint programming.

Balancing resource efficiency and fairness is critical in networked systems that support modern learning applications. We introduce the \emph{Fair Minimum Labeling} (FML) problem: the task of designing a minimum-cost temporal edge activation plan that ensures each group of nodes in a network has sufficient access to a designated target set, according to specified coverage requirements. FML captures key trade-offs in systems where edge activations incur resource costs and equitable access is essential, such as distributed data collection, update dissemination in edge-cloud systems, and fair service restoration in critical infrastructure. We first give a structural characterisation of the single-terminal case, showing that it is equivalent to the rooted Covering Steiner problem. We prove that FML is NP-hard and admits no $((1-ε)\ln |\mathcal{C}|)$-approximation for $|\mathcal{C}|$ groups, already on a star, while for any fixed number of groups it inherits a constant-factor approximation and remains APX-hard. We then present probabilistic approximation algorithms for the two-group, single-terminal case: an algorithm whose tree subroutine is exact, hence optimal on tree-structured networks and $\mathcal{O}(\log |V|)$ in expectation on general graphs, together with a faster bicriteria variant whose coverage violation degrades gracefully with the merge depth of the tree computation. For practical scalability, we additionally introduce a graph-native variant based on a shortest-path-tree reduction. Empirical results show that FML enforces group-level fairness, while the graph-native variant substantially improves scalability and achieves competitive activation cost.

The pretrain-transfer paradigm, which underpins the success of large language models (LLMs), has demonstrated the immense power of creating foundation models that learn generalizable representations from vast datasets. However, extending this paradigm to Operations Research (OR) problems on graph structures remains challenging due to the fundamental conflict between the statistical flexibility of language and the strict combinatorial constraints of graphs. To bridge this gap, we introduce the Graph Foundation Model (GFM), the first framework capable of solving all distance-based optimization problems on graph structures. By introducing the LLM-like self-supervised pre-training paradigm on the paths generated from random walks in the graph, GFM is compelled to internalize the graph's complex topological and combinatorial rules, where the connectivity of the structure itself can be treated as the supervisory signal. Unlike existing neural methods that learn complex and task-specific solving policies, our approach leverages the pre-trained GFM as a foundational model of the graph's intrinsic structure, which in turn enables a simple generative heuristic to tackle a diverse range of optimization challenges effectively. Comprehensive experiments on networks ranging from 20 to 893 nodes demonstrate that GFM achieves competitive performance against specialized solvers across a variety of distinct optimization task classes, while maintaining significantly faster inference times. Our work establishes a new paradigm of adapting the pretrain-transfer framework to graph optimization, opening the door for applying foundation model innovations to OR.

Mixed bundle pricing is a classic revenue management problem arising in industries such as e-commerce, tourism, and video games. It refers to designing product combinations (i.e., bundles) and determining their prices to maximize expected profit. Exact mixed-bundling formulations capture this structure but are computationally intractable because the number of possible bundles grows exponentially with the number of products. We propose a graph neural network (GNN)-guided pruning framework for scalable (non-)additive bundle pricing. Instead of learning on the exponential bundle-level formulation, we encode each instance as a compact customer-product graph and train an edge-output GNN to learn the product-assignment probabilities from optimal mixed-bundling solutions. The predicted probabilities are then converted into restricted candidate bundle families through fixed cutoff pruning and progressive cutoff pruning; the final prices and assignments are obtained by solving the mixed bundling formulation over the retained bundles. We further introduce a GNN-guided local search and an iterative self-improvement procedure for larger instances. The local search refines the retained bundle family by prioritizing high-confidence add/drop moves, while the iterative self-improvement procedure generates high-quality solutions on larger instances for retraining. Theoretically, we show that under mild distinguishability conditions the proposed edge-output GNN class is expressive enough to recover the optimal product-assignment mapping. Experiments show that the proposed policies recover over 98% of the optimal profit on small instances and outperform bundle-size pricing on larger instances with substantial runtime savings.

The primary paradigm in Neural Combinatorial Optimization (NCO) consists of construction methods, where a neural network is trained to sequentially add one solution component at a time until a complete solution is formed. We observe that the typical changes to the state between two steps are small, since usually only the node added to the solution is removed from the state. An efficient model should be able to reuse computation from prior steps. To that end, we propose a recurrent encoder that computes state embeddings based not only on the current state but also on embeddings from the previous state. We show that this recurrent encoder can achieve equivalent or better performance than a non-recurrent encoder even with $3\times$ fewer layers, thus significantly improving latency. We demonstrate our findings on three different problems: the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP), and the Orienteering Problem (OP), and integrate the models into a large neighborhood search algorithm to showcase the practical relevance of our findings.

Although the Maximum Clique Problem (MCP) has been extensively studied and features a rich ecosystem of exact solvers, empirical evidence shows that solver performance varies substantially across graph families. Consequently, selecting an appropriate algorithm for a given instance remains an open and practically important challenge that has received little systematic attention. We address this gap by developing an instance-aware selection framework that systematically combines global statistical descriptors with learned topological representations. We construct a comprehensive benchmark by evaluating four state-of-the-art exact solvers on a diverse collection of graph instances and deriving both global statistical and local structural features. An evaluation of conventional classifiers establishes Random Forest as a strong baseline and reveals that connectivity and topological features are key predictors of performance. Motivated by these observations, we introduce a dual-channel architecture that jointly leverages a Graph Attention Network for capturing local neighborhood patterns and a Multilayer Perceptron for modeling global statistical features. Extensive experiments show that the proposed dual-channel model consistently surpasses classical baselines and the single-best solver, achieving 90.43% test accuracy. These findings demonstrate the value of integrating local topological encoding with global statistical cues for combinatorial algorithm selection. Code and models are available at: https://anonymous.4open.science/r/GAT-MLP-7E5F.