Combinatorial Optimization

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Combinatorial optimization studies how to choose the best arrangement, assignment, route, or schedule when decisions are discrete and constraints interact. Exact methods such as integer programming, constraint programming, SAT, dynamic programming, and branch-and-bound can certify optimality, while approximation algorithms and metaheuristics trade guarantees for speed on larger instances. Learning methods—including graph neural networks, reinforcement learning, and language-model-guided heuristic design—are increasingly paired with classical search to propose solutions, warm-start solvers, or select algorithms. A central challenge is making these methods scale and generalize without sacrificing feasibility, solution quality, or the guarantees that exact and approximation methods provide.

Latest papers

386 papers47 papers in the last four weeks, up 47% on the four weeks before. 0.3% of all new papers.

Apr 13, 2026

Generating Hadamard matrices with transformers

We present a new method for constructing Hadamard matrices that combines transformer neural networks with local search in the PatternBoost framework. Our approach is designed for extremely sparse combinatorial search problems and is particularly effective for Hadamard matrices of Goethals--Seidel type, where Fourier methods permit fast scoring and optimisation. For orders between 100 and 200, it produces large numbers of inequivalent Hadamard matrices, and for larger orders, it succeeds where local search from random initialisation fails. The largest example found by our method has order 252. In addition to these new constructions, our experiments reveal that the transformer can discover and exploit useful hidden symmetry in the search space.

Geordie Williamson, Oded Yacobi, Paul Zinn-Justin

Combinatorial Optimization Neural Combinatorial Optimization

Apr 13, 2026

On the Use of Bi-Objective Evolutionary Algorithms for the Stochastic MKP under Dynamic Constraints

The multiple knapsack problem (MKP) generalizes the classical knapsack problem by assigning items to multiple knapsacks subject to capacity constraints. It is used to model many real-world resource allocation and scheduling problems. In practice, these optimization problems often involve stochastic and dynamic components. Evolutionary algorithms provide a flexible framework for addressing such problems under uncertainty and dynamic changes. In this paper, we investigate a stochastic and dynamic variant of MKP with chance constraints, where the item weights are modeled as independent normally distributed random variables and knapsack capacities change during the optimization process. We formulate the problem as a bi-objective optimization formulation that balances profit maximization and probabilistic capacity satisfaction at a given confidence level. We conduct an empirical comparison of four widely used multi-objective evolutionary algorithms (MOEAs), representing both decomposition- and dominance-based search paradigms. The algorithms are evaluated under varying uncertainty levels, confidence thresholds, and dynamic change settings. The results provide comparative insights into the behavior of decomposition-based and dominance-based MOEAs for stochastic MKP under dynamic constraints.

Ishara Hewa Pathiranage, Aneta Neumann

Machine Learning and Optimisation, School of Computer Science and Information Technology, Adelaide University, Adelaide, Australia

Combinatorial Optimization Constrained Optimization Multi-Objective Optimization Multi-Objective Evolutionary Optimization Chance-Constrained Optimization

Apr 11, 2026

Graph-RHO: Critical-path-aware Heterogeneous Graph Network for Long-Horizon Flexible Job-Shop Scheduling

Long-horizon Flexible Job-Shop Scheduling~(FJSP) presents a formidable combinatorial challenge due to complex, interdependent decisions spanning extended time horizons. While learning-based Rolling Horizon Optimization~(RHO) has emerged as a promising paradigm to accelerate solving by identifying and fixing invariant operations, its effectiveness is hindered by the structural complexity of FJSP. Existing methods often fail to capture intricate graph-structured dependencies and ignore the asymmetric costs of prediction errors, in which misclassifying critical-path operations is significantly more detrimental than misclassifying non-critical ones. Furthermore, dynamic shifts in predictive confidence during the rolling process make static pruning thresholds inadequate. To address these limitations, we propose Graph-RHO, a novel critical-path-aware graph-based RHO framework. First, we introduce a topology-aware heterogeneous graph network that encodes subproblems as operation-machine graphs with multi-relational edges, leveraging edge-feature-aware message passing to predict operation stability. Second, we incorporate a critical-path-aware mechanism that injects inductive biases during training to distinguish highly sensitive bottleneck operations from robust ones. Third, we devise an adaptive thresholding strategy that dynamically calibrates decision boundaries based on online uncertainty estimation to align model predictions with the solver's search space. Extensive experiments on standard benchmarks demonstrate that \mbox{Graph-RHO} establishes a new state of the art in solution quality and computational efficiency. Remarkably, it exhibits exceptional zero-shot generalization, reducing solve time by over 30% on large-scale instances (2000 operations) while achieving superior solution quality. Our code is available here.

Yujie Li, Jiuniu Wang, Mugen Peng +2

Beijing University of Posts and Telecommunications Beijing, China · City University of Hong Kong Hong Kong, China · Chinese Academy of Sciences Beijing, China

Flexible Job Shop Scheduling Job Shop Scheduling Combinatorial Optimization

Apr 8, 2026

Validated Intent Compilation for Constrained Routing in LEO Mega-Constellations

Operating LEO mega-constellations requires translating high-level operator intents ("reroute financial traffic away from polar links under 80 ms") into low-level routing constraints -- a task that demands both natural language understanding and network-domain expertise. We present an end-to-end system comprising three components: (1) a GNN cost-to-go router that distills Dijkstra-quality routing into a 152K-parameter graph attention network achieving 99.8% packet delivery ratio with 17x inference speedup; (2) an LLM intent compiler that converts natural language to a typed constraint intermediate representation using few-shot prompting with a verifier-feedback repair loop, achieving 98.4% compilation rate and 87.6% full semantic match on feasible intents in a 240-intent benchmark (193 feasible, 47 infeasible); and (3) an 8-pass deterministic validator with constructive feasibility certification that achieves 0% unsafe acceptance on all 47 infeasible intents (30 labeled + 17 discovered by Pass 8), with 100% corruption detection across 240 structural corruption tests and 100% on 15 targeted adversarial attacks. End-to-end evaluation across four constrained routing scenarios confirms zero constraint violations with both routers. We further demonstrate that apparent performance gaps in polar-avoidance scenarios are largely explained by topological reachability ceilings rather than routing quality, and that the LLM compiler outperforms a rule-based baseline by 46.2 percentage points on compositional intents. Our system bridges the semantic gap between operator intent and network configuration while maintaining the safety guarantees required for operational deployment.

Yuanhang Li

Intent-Based Networking LEO Satellite Networks Graph Neural Networks Combinatorial Optimization

Apr 8, 2026

Block-Bench: A Framework for Controllable and Transparent Discrete Optimization Benchmarking

We present a novel approach for constructing discrete optimization benchmarks that enables fine-grained control over problem properties, and such benchmarks can facilitate analyzing discrete algorithm behaviors. We build benchmark problems based on a set of block functions, where each block function maps a subset of variables to a real value. Problems are instantiated through a set of block functions, weight factors, and an adjacency graph representing the dependency among the block functions. Through analyzing intermediate block values, our framework allows to analyze algorithm behavior not only in the objective space but also at the level of variable representations in the obtained solutions. This capacity is particularly useful for analyzing discrete heuristics in large-scale multi-modal problems, thereby enhancing the practical relevance of benchmark studies. We demonstrate how the proposed approach can inspire the related work in self-adaptation and diversity control in evolutionary algorithms. Moreover, we explain that the proposed benchmark design enables explicit control over problem properties, supporting research in broader domains such as dynamic algorithm configuration and multi-objective optimization.

Furong Ye, Frank Neumann, Thomas Bäck +1

LIACS, Leiden University, Leiden, the Netherlands · Optimisation and Logistics, School of Computer Science and Information Technology, Adelaide University, Adelaide, Australia

Benchmark Design Synthetic Benchmark Generation Combinatorial Optimization

Apr 8, 2026

A First Guess Is Rarely the Final Answer: Learned Local Search That Generalizes to Larger Routing Problems

Neural routing solvers rarely stop at their first solution: constructed tours are usually refined by search, such as sampling or local search. Neural improvement methods learn this search as a policy that repeatedly applies local moves, but existing policies degrade sharply beyond their training size. We argue that size generalization depends on whether the information that determines the value of a move reaches the policy in a size-independent form, and identify three places where existing policies fall short: the representation of the incumbent solution, the scale of the input features, and the computation of move gains. We address all three in NICO, a learned 2-opt policy for routing that represents the incumbent by its solution edges, one token per edge without positional encodings, describes them with scale-free features, and adds the exact gain of each move to its logit, so that the network learns a correction to greedy selection. A controlled factorial study shows that edge tokens outperform node tokens with positional encodings, that the exact gains are necessary for transfer in the tested configuration, and that scale-free features help only together with them. Trained on instances with at most 100 nodes, NICO reaches a 1.1% gap on the Euclidean TSP with 1000 nodes, against 10.7% for greedy 2-opt and 36% for the best prior learned policy. It also improves the tours of constructive and diffusion-based solvers, is the best improvement method on road-network costs, and on the CVRP outperforms every classical control.

Andoni Irazusta Garmendia

University of the Basque Country (UPV/EHU)

Traveling Salesperson Problem Neural Combinatorial Optimization Combinatorial Optimization Local Search RL for Combinatorial Optimization

Apr 8, 2026

The Traveling Thief Problem with Time Windows: Benchmarks and Heuristics

While traditional optimization problems were often studied in isolation, many real-world problems today require interdependence among multiple optimization components. The traveling thief problem (TTP) is a multi-component problem that has been widely studied in the literature. In this paper, we introduce and investigate the TTP with time window constraints which provides a TTP variant highly relevant to real-world situations where good can only be collected at given time intervals. We examine adaptions of existing approaches for TTP and the Traveling Salesperson Problem (TSP) with time windows to this new problem and evaluate their performance. Furthermore, we provide a new heuristic approach for the TTP with time windows. To evaluate algorithms for TTP with time windows, we introduce new TTP benchmark instances with time windows based on TTP instances existing in the literature. Our experimental investigations evaluate the different approaches and show that the newly designed algorithm outperforms the other approaches on a wide range of benchmark instances.

Helen Yuliana Angmalisang, Frank Neumann

Optimisation and Logistics School of Computer Science and Information Technology Adelaide University Adelaide, SA 5000

Combinatorial Optimization

Apr 7, 2026

Cayley Graph Optimization for Scalable Multi-Agent Communication Topologies

Large-scale multi-agent communication has long faced a scalability bottleneck: fully connected networks require quadratic complexity, yet existing sparse topologies rely on hand-crafted rules. This paper treats the communication graph itself as a design variable and proposes CayleyTopo, a family of circulant Cayley graphs whose generator sets are optimized to minimize diameter, directly targeting worst-case information propagation speed. To navigate the enormous search space of possible generator sets, we develop a lightweight reinforcement learning framework that injects a number-theoretic prior to favor structurally rich generators, alongside a message-propagation score that provides dense connectivity feedback during construction. The resulting CayleyTopo consistently outperforms existing hand-crafted topologies, achieving faster information dissemination, greater resilience to link failures, and lower communication load, all while approaching the theoretical Moore bound. Our study opens the door to scalable, robust, and efficient communication foundations for future multi-agent systems, where the graph itself becomes optimizable rather than a fixed constraint.

Jingkai Luo, Yulin Shao

Department of Electrical and Computer Engineering, The University of Hong Kong, Hong Kong, China

Communication Topology Optimization Multi-Agent Communication Topology Multi-Agent Communication Combinatorial Optimization

Apr 7, 2026

Inventory of the 12 007 Low-Dimensional Pseudo-Boolean Landscapes Invariant to Rank, Translation, and Rotation

Many randomized optimization algorithms are rank-invariant, relying solely on the relative ordering of solutions rather than absolute fitness values. We introduce a stronger notion of rank landscape invariance: two problems are equivalent if their ranking, but also their neighborhood structure and symmetries (translation and rotation), induce identical landscapes. This motivates the study of rank landscapes rather than individual functions. While prior work analyzed the rankings of injective function classes in isolation, we provide an exhaustive inventory of the invariant landscape classes for pseudo-Boolean functions of dimensions 1, 2, and 3, including non-injective cases. Our analysis reveals 12,007 classes in total, a significant reduction compared to rank-invariance alone. We find that non-injective functions yield far more invariant landscape classes than injective ones. In addition, complex combinations of topological landscape properties and algorithm behaviors emerge, particularly regarding deceptiveness, neutrality, and the performance of hill-climbing strategies. The inventory serves as a resource for pedagogical purposes and benchmark design, offering a foundation for constructing larger problems with controlled hardness and advancing our understanding of landscape difficulty and algorithm performance.

Arnaud Liefooghe, Sébastien Verel

LISIC · Univ. Littoral Côte d’Opale, UR 4491, LISIC, F-62100 Calais, France

Fitness Landscape Analysis Combinatorial Optimization Evolutionary Optimization

Apr 4, 2026

TransGP: Task-Conditioned Transformer-Guided Genetic Programming for Multitask Dynamic Flexible Job Shop Scheduling

Hyper-heuristics have become a popular approach for solving dynamic flexible job shop scheduling (DFJSS) problems. They use gradient-free optimization techniques like Genetic Programming (GP) to evolve non-differentiable heuristics. However, conventional GP methods tend to converge slowly because they rely solely on evolutionary search to find good heuristics. Existing multitask GP methods can solve multiple tasks simultaneously and speed up the search by transferring knowledge across similar tasks. But they mostly exchange heuristic building blocks without truly generating heuristics conditioned on task information. In this paper, we aim to accelerate convergence and enable task-specific heuristic generation by incorporating a task-conditioned Transformer model. The Transformer works in two ways. First, it learns the distribution of elite heuristics, biasing the search toward promising regions of the heuristic space. Second, through conditional generation, it produces heuristics tailored to specific tasks, allowing the model to handle multiple scheduling tasks at once and improving overall optimization efficiency. Based on these ideas, we propose TransGP, a Task-Conditioned Transformer-Guided GP framework. This evolutionary paradigm integrates generative modeling with GP, enabling efficient multitask heuristic learning and knowledge transfer. We evaluate TransGP on a range of DFJSS scenarios. Experimental results show that TransGP consistently outperforms multitask GP baselines, widely used handcrafted heuristics, and the pure Transformer model, achieving faster convergence, superior solution quality, and enhanced robustness.

Meng Xu, Jiao Liu, Hua Yu +1

Job Shop Scheduling Automated Heuristic Design Genetic Programming Combinatorial Optimization Multi-Task Optimization

Apr 3, 2026

Breaking Validity-Induced Boundaries to Expand Algorithm Search Space: A Two-Stage AST-Based Operator for LLM-Driven Automated Heuristic Evolution

Large Language Model (LLM) based automated heuristic design (AHD) has shown great potential in discovering efficient heuristics. Most existing LLM-AHD frameworks use semantic evolutionary operators that rely entirely on the LLM's pre-trained knowledge. These one-stage methods strictly require the generated code to be valid during the operation and often rely on a ``thought-code'' representation. We argue that this end-to-end generation fundamentally limits the exploration ability within the algorithm search space. In this paper, we propose a two-stage, structure-based evolutionary operator for LLM-AHD. In the first stage, our approach directly performs crossover and mutation on the Abstract Syntax Trees (ASTs) of the heuristic code, intentionally generating diverse but often invalid structural variants. In the second stage, the LLM is employed to repair these invalid heuristics into executable, high-quality code. Depending on the underlying framework, either the raw invalid variants or the repaired heuristics are integrated into the population to preserve potential structural patterns. We demonstrate that the proposed operator can significantly enhance the search ability of state-of-the-art LLM-AHD algorithms, such as EoH-S. Experimental results on the Traveling Salesman Problem (TSP) and the Online Bin Packing Problem (OBP) show that our method effectively improves both optimization performance and convergence speed.

Sun Shengming, Shi Jialong

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China

Automated Heuristic Design Large Language Model-Guided Optimization Genetic Programming Combinatorial Optimization

Apr 2, 2026

How to measure the optimality of word or gesture order with respect to the principle of swap distance minimization

The structure of all the permutations of a sequence can be represented as a permutohedron, a graph where vertices are permutations and two vertices are linked if a swap of adjacent elements in the permutation of one of the vertices produces the permutation of the other vertex. It has been hypothesized that word orders in languages minimize the swap distance in the permutohedron: given a source order, word orders that are closer in the permutohedron should be less costly and thus more likely. Here we explain how to measure the degree of optimality of word order variation with respect to swap distance minimization. We illustrate the power of our novel mathematical framework by examining spontaneous gestures produced by speakers from four languages (two SVO languages and two VSO languages, which represent two linguistic families). We show that crosslinguistic gestures are at least

77\%

optimal. It is unlikely that the multiple times where crosslinguistic gestures hit optimality are due to chance. We establish the theoretical foundations for research on the optimality of word or gesture order with respect to swap distance minimization in communication systems. Finally, we introduce the quadratic assignment problem (QAP) into language research as an umbrella for multiple optimization problems and, accordingly, postulate a general principle of optimal assignment that unifies various linguistic principles including swap distance minimization.

Ramon Ferrer-i-Cancho

Quantitative, Mathematical and Computational Linguistics Research Group (LQMC), Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Campus Nord, Edifici Omega, Jordi Girona Salgado 1-3, 08034 Barcelona, Catalonia, Spain

Combinatorial Optimization

Apr 2, 2026

When does learning pay off? A study on DRL-based dynamic algorithm configuration for carbon-aware scheduling

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.

Andrea Mencaroni, Robbert Reijnen, Yingqian Zhang +1

Department of Industrial Systems Engineering and Product Design, Ghent University, Ghent, Belgium · Industrial Systems Engineering (ISyE), Flanders Make, Kortrijk, Belgium · Eindhoven Artificial Intelligence Systems Institute, Eindhoven University of Technology, The Netherlands +1

RL Control Learning to Optimize Combinatorial Optimization

Apr 2, 2026

Solving the Two-dimensional single stock size Cutting Stock Problem with SAT and MaxSAT

Cutting rectangular items from stock sheets to satisfy demands while minimizing waste is a central manufacturing task. The Two-Dimensional Single Stock Size Cutting Stock Problem (2D-CSSP) generalizes bin packing by requiring multiple copies of each item type, which causes a strong combinatorial blow-up. We present a SAT-based framework where item types are expanded by demand, each copy has a sheet-assignment variable and non-overlap constraints are activated only for copies assigned to the same sheet. We also introduce an infeasible-orientation elimination rule that fixes rotation variables when only one orientation can fit the sheet. For minimizing the number of sheets, we compare three approaches: non-incremental SAT with binary search, incremental SAT with clause reuse across iterations and weighted partial MaxSAT. On the Cui--Zhao benchmark suite, our best SAT configurations certify two to three times more instances as provably optimal and achieve lower optimality gaps than OR-Tools, CPLEX and Gurobi. The relative ranking among SAT approaches depends on rotation: incremental SAT is strongest without rotation, while non-incremental SAT is more effective when rotation increases formula size.

Tuyen Van Kieu, Chi Linh Hoang, Khanh Van To

VNU University of Engineering and Technology, Hanoi, Vietnam

Boolean Satisfiability Combinatorial Optimization

Apr 2, 2026

A Graph Neural Network Approach for Solving the Ranked Assignment Problem in Multi-Object Tracking

Associating measurements with tracks is a crucial step in Multi-Object Tracking (MOT) to guarantee the safety of autonomous vehicles. To manage the exponentially growing number of track hypotheses, truncation becomes necessary. In the

δ

-Generalized Labeled Multi-Bernoulli (

δ

-GLMB) filter application, this truncation typically involves the ranked assignment problem, solved by Murty's algorithm or the Gibbs sampling approach, both with limitations in terms of complexity or accuracy, respectively. With the motivation to improve these limitations, this paper addresses the ranked assignment problem arising from data association tasks with an approach that employs Graph Neural Networks (GNNs). The proposed Ranked Assignment Prediction Graph Neural Network (RAPNet) uses bipartite graphs to model the problem, harnessing the computational capabilities of deep learning. The conclusive evaluation compares the RAPNet with Murty's algorithm and the Gibbs sampler, showing accuracy improvements compared to the Gibbs sampler.

Robin Dehler, Martin Herrmann, Jan Strohbeck +1

The authors are with the Institute of Measurement, Control and Microtechnology, Ulm University, D-89081 Ulm, Germany.

Data Association Multi-Object Tracking Combinatorial Optimization Graph Neural Networks

Apr 1, 2026

Infeasibility Aware Large Language Models for Combinatorial Optimization

Large language models (LLMs) are increasingly explored for NP-hard combinatorial optimization problems, but most existing methods emphasize feasible-instance solution generation and do not explicitly address infeasibility detection. We propose an infeasibility-aware framework that combines certifiable dataset construction, supervised fine-tuning, and LLM-assisted downstream search. For the minor-embedding problem, we introduce a new mathematical programming formulation together with provable zero-phase infeasibility screening, which enables scalable construction of training instances labeled either as feasible with structured certificates or as certifiably infeasible. Using training data generated through this exact optimization pipeline, we show that an 8B-parameter LLM can be fine-tuned to jointly perform solution generation and infeasibility detection. We further utilize LLM outputs as warm starts for downstream local search, providing a practical way to accelerate optimization even when the LLM outputs are imperfect. Experiments show that our fine-tuned model improves overall accuracy by up to 30% over GPT-5.2; meanwhile LLM-guided warm starts provide up to

2\times

speedup compared with starting from scratch in downstream local search.

Yakun Wang, Min Chen, Zeguan Wu +3

Lehigh University, Bethlehem, PA 18015, USA · Department of Computer Science, University of Pittsburgh, Pittsburgh, PA 15260, USA · Johnson & Johnson, New Brunswick, NJ 08933, USA

Combinatorial Optimization Large Language Model-Guided Optimization LLM Fine-Tuning

Mar 31, 2026

Optimizing Donor Outreach for Blood Collection Sessions: A Scalable Decision Support Framework

Blood donation centers face challenges in matching supply with demand while managing donor availability. Although targeted outreach is important, it can cause donor fatigue via over-solicitation. Effective recruitment requires targeting the right donors at the right time, balancing constraints with donor convenience and eligibility. Despite extensive work on blood supply chain optimization and growing interest in algorithmic donor recruitment, the operational problem of assigning donors to sessions across a multi-site network, taking into account eligibility, capacity, blood-type demand targets, geographic convenience, and donor safety, remains unaddressed. We address this gap with an optimization framework for donor invitation scheduling incorporating donor eligibility, travel convenience, blood-type demand targets, and penalties. We evaluate two strategies: (i) a binary integer linear programming (BILP) formulation and (ii) an efficient greedy heuristic. Evaluation uses the registry from Instituto Português do Sangue e da Transplantação (IPST) for invite planning in the Lisbon operational region using 4-month windows. A prospective pipeline integrates organic attendance forecasting, quantile-based demand targets, and residual capacity estimation for forward-looking invitation plans. Results reveal its key role in closing the supply-demand gap in the Lisbon operational region. A controlled comparison shows that the greedy heuristic achieves results comparable to the BILP, with 188x less peak memory and 115x faster runtime; trade-offs include 3.9 pp lower demand fulfillment (86.1% vs. 90.0%), larger donor-session distance, higher adverse-reaction donor exposure, and greater invitation burden per non-high-frequency donor, reflecting local versus global optimization. Experiments assess how constraint-aware scheduling can close gaps by mobilizing eligible inactive/lapsing donors.

André Carneiro, Pedro T. Monteiro, Rui Henriques

INESC-ID and Técnico Universidade de Lisboa Portugal

Healthcare Supply Chain Management Combinatorial Optimization

Mar 31, 2026

Reducing Complexity for Quantum Approaches in Train Load Optimization

Efficiently planning container loads onto trains is a computationally challenging combinatorial optimization problem, central to logistics and supply chain management. A primary source of this complexity arises from the need to model and reduce rehandle operations-unproductive crane moves required to access blocked containers. Conventional mathematical formulations address this by introducing explicit binary variables and a web of logical constraints for each potential rehandle, resulting in large-scale models that are difficult to solve. This paper presents a fundamental departure from this paradigm. We introduce an innovative and compact mathematical formulation for the Train Load Optimization (TLO) problem where the rehandle cost is calculated implicitly within the objective function. This novel approach helps prevent the need for dedicated rehandle variables and their associated constraints, leading to a dramatic reduction in model size. We provide a formal comparison against a conventional model to analytically demonstrate the significant reduction in the number of variables and constraints. The efficacy of our compact formulation is assessed through a simulated annealing metaheuristic, which finds high-quality loading plans for various problem instances. The results confirm that our model is not only more parsimonious but also practically effective, offering a scalable and powerful tool for modern rail logistics.

Zhijie Tang, Albert Nieto-Morales, Arit Kumar Bishwas

Commercial Technology and Innovation, Shanghai, China · Tech and Innovation, San Francisco, US

Combinatorial Optimization Supply Chain Management

Mar 30, 2026

Differentiable Initialization-Accelerated CPU-GPU Hybrid Combinatorial Scheduling

This paper presents a hybrid CPU-GPU framework for solving combinatorial scheduling problems formulated as Integer Linear Programming (ILP). While scheduling underpins many optimization tasks in computing systems, solving these problems optimally at scale remains a long-standing challenge due to their NP-hard nature. We introduce a novel approach that combines differentiable optimization with classical ILP solving. Specifically, we utilize differentiable presolving to rapidly generate high-quality partial solutions, which serve as warm-starts for commercial ILP solvers (CPLEX, Gurobi) and rising open-source solver HiGHS. This method enables significantly improved early pruning compared to state-of-the-art standalone solvers. Empirical results across industry-scale benchmarks demonstrate up to a

10\times

performance gain over baselines, narrowing the optimality gap to

<0.1\%

. This work represents the first demonstration of utilizing differentiable optimization to initialize exact ILP solvers for combinatorial scheduling, opening new opportunities to integrate machine learning infrastructure with classical exact optimization methods across broader domains.

Mingju Liu, Jiaqi Yin, Alvaro Velasquez +1

University of Maryland, College Park, Maryland, USA · University of Colorado, Boulder, Colorado, USA

Combinatorial Optimization Integer Linear Programming Differentiable Optimization

Mar 30, 2026

Transferable knowledge graphs with executable learned operators for algorithm design

Procedural knowledge in algorithm design is embedded in source code and rebuilt for each new domain. We introduce Generative Executable Algorithm Knowledge Graphs (GEAKG), a representation in which this knowledge is stored as a generative, executable, transferable graph: typed nodes hold validated operators, edges encode admissible compositions, and learned edge weights record effective sequences. The same engine instantiates the structure across domains by changing only a role ontology (RoleSchema) and a binding. We study GEAKG as a representation mechanism rather than a state-of-the-art optimizer, asking what transfers and when. Layer ablations localize transfer by granularity: within a neural-architecture-search family the learned snapshot transfers across 70 dataset pairs - its weights stay correlated across datasets and one frozen snapshot remains competitive with Regularized Evolution at zero deployment-token cost; across combinatorial domains only the ontology-constrained executable structure transfers, not the learned weights. That structure pays off where target-side search is expensive - a Traveling Salesman snapshot beats an equally untuned from-scratch search on large scheduling instances even at one-fifth its budget - but does not improve on an effective local search where one is cheap, as in assignment and linear ordering. Executable procedural knowledge can thus be acquired offline, compacted, inspected, and reused without runtime language-model calls.

Camilo Chacón Sartori, José H. García, Andrei Voicu Tomut +1

Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193, Barcelona, Spain · Artificial Intelligence Research Institute (IIIA-CSIC), Bellaterra, Spain

Knowledge Graphs Automated Algorithm Discovery Automated Heuristic Design KG Construction Combinatorial Optimization

Mar 29, 2026

DSevolve: Enabling Real-Time Adaptive Scheduling on Dynamic Flexible Job Shop with LLM-Evolved Heuristic Portfolios

In dynamic flexible job shops, order arrivals, machine breakdowns, and processing-time deviations continually reshape the scheduling state and the priority trade-offs behind dispatching decisions. Dispatching rules are well suited to this setting because they are fast, interpretable, and easy to deploy, and recent LLM-assisted automatic heuristic design further expands their expressiveness by evolving composite priority functions. The key challenge is to make these evolved rule behaviors state-adaptive without losing the rapid response needed for online rescheduling. This paper proposes a dynamic self-evolutionary framework DSevolve, which separates offline rule-library construction from online state-conditioned rule selection. Offline, an LLM-guided quality-diversity search combines multi-persona seeding, a MAP-Elites behavioral archive, and behavior-guided variation to evolve a library of complementary rules rather than a single elite, and event-level simulation then trains a neural selector to rank the rules by state. Online, after each rescheduling event, a neural selector maps a 22-dimensional state fingerprint to rule scores and dispatches the top-ranked rule within about a second, meeting the response-time requirement after each disruption. Experiments on dynamic instances derived from standard flexible job shop benchmarks show that DSevolve achieves lower mean makespan than individual LLM-evolved rules, classical dispatching rules, and learning-based baselines under a one-active-rule deployment protocol. Trained only on small instances, the selector transfers zero-shot to substantially larger dynamic shops. These results show that state-conditioned selection preserves the speed and interpretability of dispatching rules while improving adaptability.

XinLei Zhou, Jin Huang, Jie Yang +2

School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, PR China

Flexible Job Shop Scheduling Real-Time Scheduling Combinatorial Optimization Automated Heuristic Design Metaheuristic Optimization

Mar 29, 2026

A gentle tutorial on Bock's algorithm for minimum directed spanning trees with a structured reformulation

Bock's 1971 algorithm is an exact primal--dual method for the minimum-cost arborescence problem, but its Algol presentation obscures the interaction of its maintained arrays and label-directed control flow. We provide a self-contained tutorial comprising the original listing, a line-mapped explanation, a circuit-forming three-node example, and a complete trace of Bock's ten-node instance. We also present a structured reformulation that replaces temporary span-label changes with explicit component and trace state. Local tightness and contraction-progress results, together with an operational-correspondence theorem, establish that the reformulation preserves Bock's candidate choices, transfers, and final solution.

Yuxi Wang, Jungyeul Park

UBC, Canada · KAIST, South Korea

Combinatorial Optimization Primal-Dual Optimization

Mar 28, 2026

Aligning LLMs with Graph Neural Solvers for Combinatorial Optimization

Recent research has demonstrated the effectiveness of large language models (LLMs) in solving combinatorial optimization problems (COPs) by representing tasks and instances in natural language. However, purely language-based approaches struggle to accurately capture complex relational structures inherent in many COPs, rendering them less effective at addressing medium-sized or larger instances. To address these limitations, we propose AlignOPT, a novel approach that aligns LLMs with graph neural solvers to learn a more generalizable neural COP heuristic. Specifically, AlignOPT leverages the semantic understanding capabilities of LLMs to encode textual descriptions of COPs and their instances, while concurrently exploiting graph neural solvers to explicitly model the underlying graph structures of COP instances. Our approach facilitates a robust integration and alignment between linguistic semantics and structural representations, enabling more accurate and scalable COP solutions. Experimental results demonstrate that AlignOPT achieves state-of-the-art results across diverse COPs, underscoring its effectiveness in aligning semantic and structural representations. In particular, AlignOPT demonstrates strong generalization, effectively extending to previously unseen COP instances.

Shaodi Feng, Zhuoyi Lin, Yaoxin Wu +4

National Yang Ming Chiao Tung University · Institute for Infocomm Research (I2R), A*STAR · Eindhoven University of Technology Eindhoven +3

Combinatorial Optimization Neural Combinatorial Optimization Large Language Model-Guided Optimization

Mar 27, 2026

On the Complexity of Optimal Graph Rewiring for Oversmoothing and Oversquashing in Graph Neural Networks

Graph Neural Networks (GNNs) face two fundamental challenges when scaled to deep architectures: oversmoothing, where node representations converge to indistinguishable vectors, and oversquashing, where information from distant nodes fails to propagate through bottlenecks. Both phenomena are intimately tied to the underlying graph structure, raising a natural question: can we optimize the graph topology to mitigate these issues? This paper provides a theoretical investigation of the computational complexity of such graph structure optimization. We formulate oversmoothing and oversquashing mitigation as graph optimization problems based on spectral gap and conductance, respectively. We prove that exact optimization for either problem is NP-hard through reductions from Minimum Bisection, establishing NP-completeness of the decision versions. Our results provide theoretical foundations for understanding the fundamental limits of graph rewiring for GNN optimization and justify the use of approximation algorithms and heuristic methods in practice.

Mostafa Haghir Chehreghani

Department of Computer Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

Over-Squashing in GNNs Over-Smoothing in GNNs Graph Neural Networks Graph Rewiring GNN Oversmoothing Combinatorial Optimization

Mar 26, 2026

Online Learning for Dynamic Constellation Topologies

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.

João Norberto, Ricardo Ferreira, Cláudia Soares

NOVA School of Science and Technology, Caparica, Portugal

Combinatorial Optimization Wireless Communications Online Learning

Mar 26, 2026

Scheduling with Time Dependent Utilities: Fairness and Efficiency

A new class of multi agent single machine scheduling problems is introduced, where each job is associated with a self interested agent with a utility function decreasing in completion time. We aim to achieve a fair solution by maximizing the minimum utility across all agents. We study the problem's complexity and propose solution methods for several variants. For the general case, we present a binary search procedure to find the largest possible minimum utility, as well as an exact greedy based alternative. Variants with release and due dates are analyzed, showing strong NP hardness for arbitrary release dates, but weak NP hardness for a single release date job, and polynomial solvability when all jobs share processing times. For all these cases we also study the corresponding problem of finding efficient solutions where the sum of utilities is maximized. We also examine settings where linear utility functions can be adjusted within budget constraints, exploring the impact on optimal schedules when intercepts or slopes are modified. From a single agent perspective, we investigate the effect of improving one agent's utility in the overall solution. Adding a new job to be inserted with the best possible utility gives rise to rescheduling problems, where different lower bounds depending on the utilities of the original fair schedule are imposed. Finally, we consider a bi level setting where a leader wants to enforce a certain target schedule by modifying utility functions while the follower computes a fair solution for the modified instance. Our work contributes to scheduling theory, multi agent systems, and algorithmic fairness, highlighting fairness oriented objectives in competitive scheduling.

Gaia Nicosia, Andrea Pacifici, Ulrich Pferschy

Dipartimento di Ingegneria Civile, Informatica e delle Tecnologie Aeronautiche, Università degli Studi “Roma Tre”, via della Vasca Navale 79, 00146 Rome, Italy · Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università degli Studi di Roma “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy · Department of Operations and Information Systems, University of Graz, Universitaetsstrasse 15, 8010 Graz, Austria

Fair Resource Allocation Combinatorial Optimization Algorithmic Fairness Multi-Agent Systems

Mar 25, 2026

A Deep Reinforcement Learning (DRL)-Based Transformer Method for Solving the Open Shop Scheduling Problem

The open shop scheduling problem (OSSP) arises in many industrial and service settings but remains computationally challenging as the number of jobs and machines increases. While exact methods quickly become intractable, classical dispatching rules and metaheuristics may require substantial tuning to maintain solution quality at large scales. This study develops a Transformer-based scheduling policy for OSSP using an encoder-decoder architecture with multi-head attention. The model is trained on Taillard benchmark instances (4x4, 5x5, 7x7, and 10x10) using only the processing-time matrix as input and produces feasible schedules with makespans typically within 15-30% of best-known values. To evaluate scalability, the trained policy is applied without retraining to randomly generated instances from 40x40 to 100x100 and compared against classical dispatching heuristics, including SPT, LPT, MWKR, and EST. Across these large instances, the Transformer achieved average gaps of 12.89-15.12% relative to a standard lower bound. Compared with EST, the Transformer remained competitive, typically within a modest margin, while substantially outperforming SPT and LPT. These results indicate that a Transformer policy trained on small OSSP instances can generalize to substantially larger problems and provide a feature-light, learning-based alternative to classical dispatching rules.

Faezeh Ardali, Mwembezi A. Nyelele, Gerald M. Knapp

Louisiana State University, Baton Rouge, LA, USA · University of Minnesota Duluth, Duluth, MN, USA

Combinatorial Optimization Neural Combinatorial Optimization RL for Combinatorial Optimization Transformer-Based RL

Mar 24, 2026

Fleet-Level Battery-Health-Aware Scheduling for Autonomous Mobile Robots

Autonomous mobile robot fleets must coordinate task allocation and charging under limited shared resources, yet most battery aware planning methods address only a single robot. This paper extends degradation cost aware task planning to a multi robot setting by jointly optimizing task assignment, service sequencing, optional charging decisions, charging mode selection, and charger access while balancing degradation across the fleet. The formulation relies on reduced form degradation proxies grounded in the empirical battery aging literature, capturing both charging mode dependent wear and idle state of charge dependent aging; the bilinear idle aging term is linearized through a disaggregated piecewise McCormick formulation. Tight big M values derived from instance data strengthen the LP relaxation. To manage scalability, we propose a hierarchical matheuristic in which a fleet level master problem coordinates assignments, routes, and charger usage, while robot level subproblems whose integer part decomposes into trivially small independent partition selection problems compute route conditioned degradation schedules. Systematic experiments compare the proposed method against three baselines: a rule based nearest available dispatcher, an energy aware formulation that enforces battery feasibility without modeling degradation, and a charger unaware formulation that accounts for degradation but ignores shared charger capacity limits.

Jiachen Li, Shihao Li, Jian Chu +2

University of Texas at Austin, Austin, TX 78712 USA

Multi-Robot Task Allocation Multi-Agent System Optimization Combinatorial Optimization

Mar 19, 2026

cuGenOpt: A GPU-Accelerated General-Purpose Metaheuristic Framework for Combinatorial Optimization

Combinatorial optimization problems arise in logistics, scheduling, and resource allocation, yet existing approaches face a fundamental trade-off among generality, performance, and usability. We present cuGenOpt, a GPU-accelerated general-purpose metaheuristic framework that addresses all three dimensions simultaneously. At the engine level, cuGenOpt adopts a "one block evolves one solution" CUDA architecture with a unified encoding abstraction (permutation, binary, integer), a two-level adaptive operator selection mechanism, and hardware-aware resource management. At the extensibility level, a user-defined operator registration interface allows domain experts to inject problem-specific CUDA search operators. At the usability level, a JIT compilation pipeline exposes the framework as a pure-Python API, and an LLM-based modeling assistant converts natural-language problem descriptions into executable solver code. Experiments across five thematic suites on three GPU architectures (T4, V100, A800) show that cuGenOpt outperforms general MIP solvers by orders of magnitude, achieves competitive quality against specialized solvers on instances up to n=150, and attains 4.73% gap on TSP-442 within 30s. Twelve problem types spanning five encoding variants are solved to optimality. Framework-level optimizations cumulatively reduce pcb442 gap from 36% to 4.73% and boost VRPTW throughput by 75-81%. Code: https://github.com/L-yang-yang/cugenopt

Yuyang Liu

Independent Researcher, Shenzhen, China

Metaheuristic Optimization Combinatorial Optimization GPU Acceleration

Mar 17, 2026

Domain-Independent Dynamic Programming with Constraint Propagation

There are two prevalent model-based paradigms for combinatorial problems: 1) state-based representations, such as heuristic search, dynamic programming (DP), and decision diagrams, and 2) constraint and domain-based representations, such as constraint programming (CP), (mixed-)integer programming, and Boolean satisfiability. In this paper, we bridge the gap between the DP and CP paradigms by integrating constraint propagation into DP, enabling a DP solver to prune states and transitions using constraint propagation. To this end, we implement constraint propagation using a general-purpose CP solver in the Domain-Independent Dynamic Programming framework and evaluate using heuristic search on three combinatorial optimisation problems: Single Machine Scheduling with Time Windows, the Resource Constrained Project Scheduling Problem (RCPSP), and the Travelling Salesperson Problem with Time Windows (TSPTW). Our evaluation shows that constraint propagation significantly reduces the number of state expansions, causing our approach to solve more instances than a DP solver for Single Machine Scheduling and RCPSP, and showing similar improvements for tightly constrained TSPTW instances. The runtime performance indicates that the benefits of propagation outweigh the overhead for constrained instances, but that further work into reducing propagation overhead could improve performance further. Our work is a key step in understanding the value of constraint propagation in DP solvers, providing a model-based approach to integrating DP and CP.

Imko Marijnissen, J. Christopher Beck, Emir Demirović +1

Delft University of Technology, The Netherlands · University of Toronto, Canada · National Institute of Informatics, Japan +1

Resource-Constrained Project Scheduling Constraint Propagation Dynamic Programming Combinatorial Optimization Constraint Satisfaction Constraint Programming