Combinatorial Optimization

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33 papers in the last four weeks, up 83% on the four weeks before. 0.3% of all new papers.

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Latest papers 277

May 12, 2026cs.AI

Formalize, Don't Optimize: The Heuristic Trap in LLM-Generated Combinatorial Solvers

Large Language Models (LLMs) struggle to solve complex combinatorial problems through direct reasoning, so recent neuro-symbolic systems increasingly use them to synthesize executable solvers. A central design question is how the LLM should represent the solver, and whether it should also attempt to optimize search. We introduce CP-SynC-XL, a benchmark of 100 combinatorial problems (4,577 instances), and evaluate three solver-construction paradigms: native algorithmic search (Python), constraint modeling through a Python solver API (Python + OR-Tools), and declarative constraint modeling (MiniZinc + OR-Tools). We find a consistent representational divergence: Python + OR-Tools attains the highest correctness across LLMs, while MiniZinc + OR-Tools has lower absolute coverage despite using the same OR-Tools back-end. Native Python is the most likely to return a schema-valid solution that fails verification, whereas solver-backed paths preserve higher conditional fidelity. On the heuristic axis, prompting for search optimization yields only small median speed-ups (1.03-1.12x) and a strongly bimodal effect: many instances slow down, and correctness drops sharply on a long tail of problems. A paired code-level audit traces these regressions to a recurring heuristic trap. Under an efficiency-oriented prompt, the LLM may replace complete search with local approximations (Python), inject unverified bounds (Python + OR-Tools), or add redundant declarative machinery that overwhelms or over-constrains the model (MiniZinc + OR-Tools). These findings support a conservative design principle for LLM-generated combinatorial solvers: use the LLM primarily to formalize variables, constraints, and objectives for verified solvers, and separately check any LLM-authored search optimization before use.

Haoyu Wang, Yuliang Song, Tao Li +5

University of Pennsylvania · University of Toronto · Google DeepMind +1

Large Language Model-Guided Optimization LLM-Based Program Synthesis Combinatorial Optimization Natural Language to Optimization Modeling Constraint Programming

May 12, 2026quant-ph

QAP-Router: Tackling Qubit Routing as Dynamic Quadratic Assignment with Reinforcement Learning

Qubit routing is a fundamental problem in quantum compilation, known to be NP-hard. Its dynamic nature makes local routing decisions propagate and compound over time, making global efficient solutions challenging. Existing heuristic methods rely on local rules with limited lookahead, while recent learning-based approaches often treat routing as a generic sequential decision problem without fully exploiting its underlying structure. In this paper, we introduce QAP-Router, framing qubit routing based on a dynamic Quadratic Assignment Problem (QAP) formulation. By modeling logical interactions, or quantum gates, as flow matrices and hardware topology as a distance matrix, our approach captures the interaction-distance coupling in a unified objective, which defines the reward in the reinforcement learning environment. To further exploit this structure, the policy network employs a solution-aware Transformer backbone that encodes the interaction between the flow matrix and the distance matrix into the attention mechanism. We also integrate a lookahead mechanism that blends naturally into the QAP framework, preventing myopic decisions. Extensive experiments on 1,831 real-world quantum circuits from the MQTBench, AgentQ and QUEKO datasets show that our method substantially reduces the CNOT gate count of routed circuits by 15.7%, 30.4% and 12.1%, respectively, relative to existing industry compilers.

Kien X. Nguyen, Ankit Kulshrestha, Ilya Safro +1

Department of Computer and Information Sciences, University of Delaware, USA · Quantum Lab, Fujitsu Research of America, USA

Quantum Circuit Compilation Combinatorial Optimization Neural Combinatorial Optimization

May 12, 2026stat.ML

Optimal Policy Learning under Budget and Coverage Constraints

We study optimal policy learning under combined budget and minimum coverage constraints. We show that the problem admits a knapsack-type structure and that the optimal policy can be characterized by an affine threshold rule involving both budget and coverage shadow prices. We establish that the linear programming relaxation of the combinatorial solution has an O(1) integrality gap, implying asymptotic equivalence with the optimal discrete allocation. Building on this result, we analyze two implementable approaches: a Greedy-Lagrangian (GLC) and a rank-and-cut (RC) algorithm. We show that the GLC closely approximates the optimal solution and achieves near-optimal performance in finite samples. By contrast, RC is approximately optimal whenever the coverage constraint is slack or costs are homogeneous, while misallocation arises only when cost heterogeneity interacts with a binding coverage constraint. Monte Carlo evidence supports these findings.

Giovanni Cerulli

CNR-IRCrES · National Research Council of Italy · Research Institute on Sustainable Economic Growth

Policy Optimization Combinatorial Optimization Constrained RL Constrained Optimization

May 11, 2026physics.soc-ph

Network-Based Interventions for HIV Prevention via Cascade-Aware Suppression of Transmission

Treating and preventing Human Immunodeficiency Virus (HIV) remains a critical global health challenge. While antiretroviral therapy provides a path toward viral suppression -- effectively eliminating an individual's transmission risk -- systemic resource constraints limit the reach of intervention efforts. This work addresses the strategic distribution of intensive resources among virally unsuppressed individuals to minimize the expected cascade of new infections within a transmission network. We formalize this challenge as a novel constrained optimization problem where we have resources to "treat"

k

out of a set

\mathbf{P}

of virally unsuppressed individuals, and establish its theoretical connections to existing computational literature. We then propose Cascade-Aware Suppression of Transmission (CAST), a polynomial-time

(δ, ε)

-approximation algorithm that achieves a

2\sqrt{|\mathbf{P}|}

approximation ratio by leveraging connections to the Minimum-

k

-Union (MkU) problem and Hoeffding-style concentration bounds. Extensive evaluations on real-world HIV networks demonstrate that CAST outperforms standard public health and computer science baselines. Furthermore, we show that CAST is empirically robust across diverse infectious disease networks, varied edge probability initializations, and settings involving imperfect network data.

Akseli Kangaslahti, Davin Choo, Milind Tambe +2

Harvard University · University of Witwatersrand · Wits Health Consortium +1

Healthcare Combinatorial Optimization Approximation Algorithms

May 11, 2026cs.AI

Rethinking Constraint Awareness for Efficient State Embedding of Neural Routing Solver

Heavy-Encoder-Light-Decoder (HELD) neural routing solvers have emerged as a promising paradigm due to their broad applicability across multiple vehicle routing problems (VRPs). However, they typically struggle with VRP variants with complex constraints. To address this limitation, this paper systematically revisits existing neural solvers from the perspective of the generation mechanism for state embeddings (i.e., query vector prior to compatibility calculation) during decoding. We identify that current mechanisms restrict the observation space during attention computation, introducing a key bottleneck to achieving high-quality solutions. Through detailed empirical analysis, we demonstrate the necessity of preserving a global observation space. To overcome the constraint-agnostic drawback inherent to global observation spaces, we propose a simple yet powerful Constraint-Aware Residual Modulation (CARM) module. By adaptively modulating the context embedding with constraint-relevant variables, CARM effectively enhances constraint awareness, enabling the neural solver to fully leverage the global observation space and generate an efficient state embedding. Extensive experimental results across two single-task and five multi-task neural routing solvers confirm that the CARM module consistently boosts baseline performance. Notably, solvers equipped with our CARM achieve substantial improvements in scaling to large-scale instances and in generalizing to unseen VRP variants. These findings provide valuable insights for the architectural design of neural routing solvers.

Canhong Yu, Changliang Zhou, Rongsheng Chen +2

College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China · School of Automation and Intelligent Manufacturing, Southern University of Science and Technology, Shenzhen, China · Guangdong Provincial Key Laboratory of Fully Actuated System Control Theory and Technology, Southern University of Science and Technology, Shenzhen, China +1

Neural Network Optimization Combinatorial Optimization Vehicle Routing Neural Combinatorial Optimization

May 11, 2026cs.RO

A cell-decomposition based path planner for 3D navigation in constrained workspaces

This paper proposes a cell decomposition algorithm for binary occupancy grids that ensures mutual complete visibility from each cell to at least one adjacent cell. This decomposition establishes a simplified framework for verifying path feasibility that can be easily embedded in optimization problems. To illustrate its utility, we formulate both second-order cone programs (SOCP) and their mixed-integer variant (MISOCP) within the proposed framework. Furthermore, we propose the KSP-SOCP method, which combines Yen's k-shortest path algorithm with the SOCP, achieving improved solutions compared to a standard SOCP approach while avoiding the computational burden of MISOCP. The cell decomposition algorithm, KSP-SOCP, and MISOCP approaches were evaluated in 9 city-like workspaces. The decomposition efficiently partitioned each map, enabling both optimization methods to compute feasible paths. The proposed KSP-SOCP achieved time performance comparable to the MISOCP while requiring less memory, making it highly suitable for large-scale problems.

João P. L. Morais, Luciano C. A. Pimenta, Marcelo A. Santos +1

Graduate Program in Electrical Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil · Department of Management, Information and Production Engineering, University of Bergamo, Dalmine, BG, Italy · Department of Electronic Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil

Combinatorial Optimization Motion Planning Robot Motion Planning

May 10, 2026cs.LG

Learning-Augmented Scalable Linear Assignment Problem Optimization via Neural Dual Warm-Starts

The Linear Assignment Problem is a fundamental combinatorial optimization task where classical exact solvers ensure optimality but suffer from an

\mathcal{O}(N^{3})

bottleneck, while recent neural approximations struggle with scalability and exactness. We propose a learning-augmented framework that accelerates exact solvers by predicting dual variables to warm-start the search, backed by a fallback mechanism to preserve worst-case guarantees. Central to our approach is RowDualNet, a lightweight, row-independent architecture that avoids the

\mathcal{O}(N^{2})

memory bottleneck of graph models, enabling scalable neural warm-starting up to

N=16{,}384

. Feasibility is guaranteed by construction via the Min-Trick mechanism, completely eliminating the need for costly iterative projections. Empirically, our method drastically reduces the search effort of the Jonker-Volgenant (LAPJV) algorithm, yielding robust zero-shot generalization with strict optimality and end-to-end speedups of over 2x on complex synthetic data, 1.25x on real-world tracking, and 1.5x on transportation networks.

Ilay Yavlovich, Jad Agbaria, Muhamed Mhamed +2

Department of Electrical and Computer Engineering, Technion – Israel Institute of Technology, Haifa, Israel.

Neural Network Optimization Combinatorial Optimization Learning-Augmented Algorithms

May 10, 2026cs.LG

Worst-Case Regret Bounds for Combinatorial Thompson Sampling in Sleeping Semi-Bandits

We revisit combinatorial Thompson sampling (CTS) for semi-bandits with sleeping arms, where arm availability varies over time and actions must satisfy combinatorial constraints, as in wireless mesh routing with fluctuating link availability. Despite its practical relevance, CTS has been hindered by several long-standing problems: (i) the absence of worst-case regret guarantees in the semi-bandit setting even without sleeping arms, (ii) the lack of theory under adversarially varying availability, and (iii) the consistently weak empirical performance of CTS with Gaussian priors (CTS-G). This paper resolves these long-standing issues by providing the first worst-case regret analysis of CTS-G, proving an upper bound of

\tilde{O}(m\sqrt{NT})

and a matching lower bound of

\tildeΩ(m\sqrt{NT})

. To bridge the gap between theory and practice, we further propose CL-SG, a simple CTS-G variant that samples a single shared Gaussian seed each round to coordinate exploration across arms. We show that CL-SG achieves an improved regret bound of

\tilde{O}(\sqrt{mNT})

, together with a matching lower bound

Ω(\sqrt{mNT})

. Experiments on real-world datasets demonstrate that CL-SG consistently outperforms strong baselines including CTS-G and CTS-B, and we open-source our implementation for reproducibility.

Zhiming Huang, Bingshan Hu, Jianping Pan

Paul G. Allen School of Computer Science & Engineering, University of Washington, WA, USA · Department of Computer Science, University of British Columbia, BC, Canada · Department of Computer Science, University of Victoria, BC, Canada

Thompson Sampling Multi-Armed Bandits Combinatorial Optimization Combinatorial Multi-Armed Bandits

May 9, 2026cs.RO

Omni-scale Learning-based Sequential Decision Framework for Order Fulfillment of Tote-handling Robotic Systems

Driven by the rapid expansion of e-commerce and small-batch production, the size of the intralogistics load unit of finished goods, semi-finished goods and raw materials is steadily shrinking. Totes are gradually replacing pallets as the primary handling and storage container. This shift has propelled tote-handling robotic systems to the forefront of automation order fulfillment centers. The order-fulfillment decisions of tote-handling robotic systems share a common order-tote-robot sequential decision-making nature. Existing studies primarily focus on decision mechanisms tailored to particular systems, making it difficult to generalize or transfer them to other contexts. We propose an Omni-scale Learning-based Sequential Decision Framework for Order Fulfillment of Tote-handling Robotic Systems (OLSF-TRS), a generalized and scalable sequential decision framework that combines structured combinatorial optimization with multi-agent reinforcement learning to coordinate order,tote, and robot decisions. On small-scale tote-handling robotic systems, OLSF-TRS achieves near-optimal performance with average optimality gaps below 3.5% across two distinct system configurations. In large-scale scenarios, OLSF-TRS consistently outperforms heuristic baselines across two different system types, reducing total tote movements by 8-12% and over 30% compared to SOTA rule-based approaches, while maintaining real-time responsiveness. These improvements translate into tangible operational benefits, including cost reduction, lower energy consumption, and enhanced throughput stability. The proposed framework delivers an efficient and unified order fulfillment decision-making framework for widely deployed tote-handling robotic systems,supporting high-quality order fulfillment in both e-commerce and industrial logistics sectors.

Jiaxin Liu, Peng Yang, Yuping Li +1

Institution of Data and Information, Shenzhen International Graduate School, Tsinghua University, Nanshan District, Shenzhen 518055, China

Combinatorial Optimization Robotics Multi-Robot Coordination Multi-Robot Task Allocation

May 9, 2026cs.AI

AHD Agent: Agentic Reinforcement Learning for Automatic Heuristic Design

Automatic heuristic design (AHD) has emerged as a promising paradigm for solving NP-hard combinatorial optimization problems (COPs). Recent works show that large language models (LLMs), when integrated into well-designed frameworks (i.e., LLM-AHD), can autonomously discover high-performing heuristics. However, existing LLM-AHD frameworks typically treat LLMs as passive generators within fixed workflows, where the model generates heuristics from manually designed, limited context. Such context may fail to capture state-dependent information (e.g., specific failure modes), leading to inefficient trial-and-error exploration. To overcome these limitations, we propose AHD Agent, a novel tool-integrated, multi-turn framework that empowers LLMs to proactively decide whether to generate heuristics or invoke tools to retrieve targeted evidence from the solving environment. To effectively train such a dynamic decision-making agent, we introduce an agentic reinforcement learning (RL) system, which leverages a novel environment synthesis pipeline to optimize a compact model's generalizable AHD capabilities. Experiments across eight diverse domains, including four held-out tasks, demonstrate that our 4B-parameter agent matches or surpasses state-of-the-art baselines using much larger models, while requiring significantly fewer evaluations. Model and inference scaling analysis further reveals that AHD Agent offers an effective trajectory toward truly autonomous heuristic design.

Haoze Lv, Ning Lu, Ziang Zhou +1

Guangdong Provincial Key Laboratory of Brain-Inspired Intelligent Computation, Department of Computer Science and Engineering, Southern University of Science and Technology · Zhongguancun Academy · The Hong Kong University of Science and Technology

Agentic RL Automated Heuristic Design Combinatorial Optimization LLM Agent Training RL for Tool Use

May 8, 2026cs.AI

HMACE: Heterogeneous Multi-Agent Collaborative Evolution for Combinatorial Optimization

Large Language Models have recently emerged as a promising paradigm for automated heuristic design for NP-hard combinatorial optimization problems. Despite this progress, existing LLM-based methods typically rely on monolithic workflows constrained by rigid templates, thereby restricting memory-guided exploration and triggering premature convergence to local optima. To design an autonomous and collaborative architecture, we introduce HMACE, a Heterogeneous Multi-Agent Collaborative Evolution framework that reconceptualizes heuristic search as an organizational design problem. HMACE decomposes each evolutionary generation into an autonomous, role-specialized loop with four coordinated agents: a Proposer for strategy exploration, a Generator for executable heuristic synthesis, an Evaluator for empirical assessment, and a Reflector for archive-backed memory update. By coupling behavior-aware retrieval, lightweight candidate filtering, and fitness-grounded archive updates, HMACE guides the search toward diverse and promising heuristic behaviors while avoiding redundant evaluations. Extensive evaluations on representative COPs, including TSP, Online BPP, MKP, and PFSP, show that HMACE achieves a favorable quality-efficiency trade-off compared to state-of-the-art single-agent and multi-agent baselines. In the matched LLM-driven reference comparison, HMACE achieves the lowest average gaps on TSP and Online BPP (0.464% and 0.223%, respectively), while requiring only 0.13M and 0.42M tokens for the two tasks, substantially fewer than the compared baselines.

Yuping Yan, Jirui Han, Fei Ming +2

School of Engineering Westlake University

Multi-Agent LLM Systems Multi-Agent System Optimization Automated Heuristic Design Evolutionary Optimization Large Language Model-Guided Optimization Combinatorial Optimization+1

May 8, 2026cs.LG

Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks

Exact solution of hard combinatorial optimization problems often relies on strong convex relaxations, but solving these relaxations repeatedly inside a branch-and-bound algorithm can be prohibitively expensive. Hence, we consider this challenge for Max-Cut, where branch and bound commonly uses semidefinite programming (SDP) relaxations to bound subproblems. We propose a Max-Cut-specific graph neural network that serves as a principled, lightweight neural proxy for these SDP solvers and can be plugged directly into an exact branch-and-bound framework. The proposed architecture has update steps of complexity

\mathcal{O}(n^2 + ne)

, and predicts both primal- and dual-feasible SDP solutions. The primal SDP solutions yield feasible Max-Cut solutions via the Goemans--Williamson algorithm. In addition, it is trained in a self-supervised fashion without requiring solved SDP relaxations as labels. Empirically, we show that our architecture can substantially reduce the cost of bounding in exact Max-Cut solving by up to

10.6 \times

compared with using the state-of-the-art SDP solver Mosek. Our work highlights the potential of learned, validity-preserving surrogates for accelerating exact optimization over structured convex relaxations.

Hao Chen, Chendi Qian, Christopher Morris +2

1Davidson School of Chemical Engineering, Purdue University · Faculty of Computer Science, RWTH Aachen University · 3Jacobs Technion-Cornell Institute, Cornell Tech

Semidefinite Programming Graph Neural Networks Branch and Bound Combinatorial Optimization Global Optimization Constrained Optimization

May 7, 2026cs.LG

Optimal Counterfactual Search in Tree Ensembles: A Study Across Modeling and Solution Paradigms

Trust in counterfactual explanations depends critically on whether their recommended changes are truly minimal: suboptimal explanations may vastly overshoot the actual changes needed to alter a decision, and heuristic errors can affect individuals unevenly, giving some users relevant recourse while assigning others unnecessarily costly recommendations. Consequently, we study the problem of computing optimal counterfactual explanations for tree ensembles under plausibility and actionability constraints. This is a combinatorial problem: for a fixed model, counterfactual search boils down to selecting consistent branching decisions and threshold-defined regions under a distance objective. We exploit this structure through CPCF, a constraint programming (CP) formulation in which numerical features are encoded as interval domains induced by split thresholds, while discrete features retain native finite-domain representations. This yields a compact finite-domain formulation that supports multiple distance objectives without continuous split-boundary search. We then place CPCF in a broader comparison across mathematical programming paradigms: we extend a maximum Boolean satisfiability (MaxSAT) formulation, originally designed for hard-voting random forests, to soft-voting ensembles, and compare against the current state-of-the-art mixed-integer linear programming (MILP) optimal approach. Across ten datasets and three types of tree ensembles, we analyze scalability, anytime performance, and sensitivity to distance metrics. We observe that CP achieves the best overall performance. More importantly, our results identify regimes in which the specific strengths of each paradigm make it best suited: CP is most versatile overall, MaxSAT handles hard-voting ensembles particularly well, and MILP remains competitive in amortized inference settings with a moderate number of split levels.

Awa Khouna, Youssouf Emine, Julien Ferry +1

Polytechnique Montréal · Ivey Business School

Counterfactual Explanations Mixed-Integer Linear Programming Combinatorial Optimization Maximum Satisfiability Constraint Programming

May 7, 2026cs.AI

Back to the Beginning of Heuristic Design: Bridging Code and Knowledge with LLMs

Large language models (LLMs) have recently advanced automatic heuristic design (AHD) for combinatorial optimization (CO), where candidate heuristics are iteratively proposed, evaluated, and refined. Most existing approaches search over executable programs and distill insights from execution feedback to guide later iterations. Because this process moves from low-level implementations to high-level principles, we refer to it as a bottom-up paradigm. We argue that this view is incomplete and introduce a complementary top-down perspective: knowledge becomes the primary search object and code merely instantiates and tests it, making what is learned explicit and reusable across problems and trajectories. We formalize this shift through a statistical-learning view that exposes a distortion--compression trade-off, and instantiate it in both population-based and tree-based AHD frameworks. Across CO and tasks beyond it, knowledge-first search improves discovery efficiency, transfer, and generalization, often outperforming code-centric pipelines, while combining both strategies yields further gains. Our results suggest that progress in AHD depends on iteratively constructing and evolving interpretable hypotheses that retain value beyond a single search trajectory.

Nguyen Viet Tuan Kiet, Bui Dinh Pham, Dao Van Tung +2

Hanoi University of Science and Technology, Hanoi, Vietnam · University of Sydney, Sydney, Australia

Automated Heuristic Design Large Language Model-Guided Optimization Combinatorial Optimization

May 6, 2026cs.LG

Graph Normalization: Fast Binarizing Dynamics for Differentiable MWIS

We introduce Graph Normalization (GN), a principled dynamical system on graphs that serves as a differentiable approximation engine for the NP-hard Maximum Weight Independent Set (MWIS) problem. MWIS encompasses many combinatorial challenges, including optimal assignment, scheduling, set packing, and MAP inference in discrete Markov Random Fields. Unlike Belief Propagation, we prove GN always converges to a binary indicator of a Maximum Independent Set. GN realizes a fast quasi-Newton descent through an exact Majorization-Minimization step, systematically improving the MWIS relaxed primal objective. We establish an equivalence between GN and the Replicator Dynamics of a nonlinear evolutionary game, where vertices compete for inclusion in an independent set. While a non-potential game, the GN game follows Fisher's Fundamental Theorem of Natural Selection, where the average fitness equals the MWIS primal objective and strictly increases. This connection leads to a weighted extension of the Motzkin-Straus theorem, showing MISes are in bijection with the local minima of a quadratic form over a tilted simplex. For the Assignment Problem, GN acts as a variant of the Sinkhorn algorithm that naturally converges to a hard assignment while generalizing to arbitrary constraint graphs. We demonstrate GN's performance as a fast binarization engine for the state-of-the-art Bregman-Sinkhorn relaxed MWIS solver. On real-world benchmarks with up to 1M edges, GN identifies solutions within 1% of the best known results in seconds on a CPU. GN opens new avenues for deep learning architectures requiring differentiable, "hard" decisions under constraints, with applications in structured sparse attention, dynamic network pruning, and Mixture-of-Experts. Beyond core AI, the GN framework enables end-to-end learning of constrained optimization in computer vision, computational biology, and resource allocation.

Laurent Guigues

Dynamical Systems Replicator Dynamics Combinatorial Optimization Nonconvex Optimization Differentiable Optimization

May 5, 2026math.OC

Exact and Evolutionary Algorithms for Sequential Multi-Objective Transmission Topology Planning

We study day-ahead transmission topology control for high-voltage grid operation under

N-1

security constraints. The operational task is to select, over a 24-hour horizon, a sequence of substation topologies obtained via busbar-coupler switching to relieve line overloads while limiting switching effort and topological complexity. We formulate this task as a sequential multi-objective optimization problem with four objectives used in TSO decision making: worst-case

N-1

line loading, maximum topological depth, number of topology changes, and time spent outside the reference topology. We propose an exact block algorithm that exploits the temporal structure of topology plans: consecutive hours with the same topology are represented as blocks, enabling enumeration of the complete Pareto front over the admissible set of topologies under fixed operational bounds on depth and switching. We also develop a tailored NSGA-III-based evolutionary heuristic and evaluate it against the exact front. Using real operational data from the Dutch high-voltage transmission grid operated by TenneT, the block algorithm computes the exact front for a highly congested day in under three minutes after topology-level load-flow preprocessing. The exact front reveals low-switching plans with no DC

N-1

thermal overloads that the tested evolutionary search fails to find. The proposed method, therefore, provides both a practical day-ahead decision-support tool for transmission operators and a benchmark for heuristic and learning-based topology-control methods.

Job Groeneveld, Miguel Muñoz, Jan Viebahn +1

TenneT TSO B.V., Arnhem, The Netherlands · Artelys, Paris, France · Department of Mathematics, Vrije Universiteit Amsterdam, The Netherlands

Multi-Objective Evolutionary Optimization Combinatorial Optimization Multi-Objective Optimization Power Systems

May 5, 2026cs.AI

Automated Large-scale CVRP Solver Design via LLM-assisted Flexible MCTS

Solving large-scale CVRP (LSCVRP) with hundreds to thousands of nodes remains difficult for even state-of-the-art solvers. Divide-and-conquer can scale by decomposing the instance into size-reduced subproblems, but designing decomposition logic and configuring sub-solvers is highly expertise- and labor-intensive. Large Language Models (LLMs) have emerged as promising tools for automated algorithm design. However, existing LLM-driven approaches struggle with LSCVRP primarily due to the difficulty in generating sophisticated search strategies within a limited context window. To bridge this gap, we propose the LLM-assisted Flexible Monte Carlo Tree Search (LaF-MCTS), a novel framework that automates the design of high-performance LSCVRP solvers. We develop a three-tier decision hierarchy to enable incremental design of decomposition policies and sub-solvers for LSCVRP. To enable efficient search within the algorithmic hypothesis space, we introduce semantic pruning to eliminate semantically and structurally redundant codes, and branch regrowth to regenerate codes and preserve diversity. Extensive experiments on CVRPLib demonstrate that LaF-MCTS autonomously composes and optimizes decomposition-enhanced solvers that surpasses various state-of-the-art CVRP solvers.

Tong Guo, Caishun Chen, Yew Soon Ong

College of Computing and Data Science, Nanyang Technological University, Singapore

Automated Algorithm Discovery Large Language Model-Guided Optimization Combinatorial Optimization Monte Carlo Tree Search Vehicle Routing

May 4, 2026cs.AI

Computing Thiele Rules on Interval Elections and their Generalizations

Approval-based committee voting has received significant attention in the social choice community. Among the studied rules, Thiele rules, and especially Proportional Approval Voting (PAV), stand out for desirable properties such as proportional representation, Pareto optimality, and support monotonicity. Their main drawback is that computing a Thiele outcome is NP-hard in general. A glimpse of hope comes from the fact that Thiele rules are better behaved under structured preferences. On the candidate interval (CI) domain, they are computable in polynomial time via a linear program (LP) that has a totally unimodular constraint matrix. Surprisingly, this approach fails for the related voter interval (VI) domain, and the complexity of the problem has repeatedly been posed as an open question. Our main result resolves this question: although the relevant matrix is not totally unimodular, the ``standard'' LP still admits at least one optimal integral solution, and we provide a fast algorithm for finding it. Our technique naturally extends to the voter-candidate interval (VCI) domain, also known as the 1-dimensional voter-candidate range (1D-VCR) domain, and to the linearly consistent (LC) domain, both of which generalize the candidate and voter interval domains. Although both the VCI and LC domains have been studied in social choice, their relationship was unknown. We show, through connections to graph theory, that LC strictly contains VCI. We also provide an alternative definition of LC that is closer in spirit to VCI and has a natural interpretation in approval elections; this equivalence may be of independent interest. Finally, we study an alternative tree-based generalization of VCI and show that Thiele rules become NP-hard to compute on this domain.

Dimitris Avramidis, Alexandra Lassota, Ulrike Schmidt-Kraepelin +1

Department of Mathematics and Computer Science Eindhoven University of Technology Eindhoven, Netherlands · School of Computer Science, and Department of Mathematics and Statistics McGill University Montréal, Canada

Social Choice Theory Combinatorial Optimization Computational Social Science

May 2, 2026cs.LG

Hybrid Quantum Reinforcement Learning with QAOA for Improved Vehicle Routing Optimization

Vehicle Routing Problem (VRP) is one of the most complex NP-hard combinatorial optimization problem in transportation and logistics that requires a dynamic solution approach. In this paper we present a new hybrid approach that combines the Quantum Approximate Optimization Algorithm (QAOA) into the QRL policy network, instead of the usual variational layers, QAOA mixing and cost Hamiltonian layers. This enhancement enables the agent to exploit problem specific particular quantum correlations when learning policies, and so richer exploration of the routing solution space. The QAOA-augmented QRL framework shows quicker convergence in training and can tackle larger VRP instances that are beyond the reach of Grover's Adaptive Search (GAS) and Quantum Reinforcement Learning (QRL) approaches. Experiments on standard VRP instances demonstrate better solutions, fewer episodes to converge and good memory usage on near term quantum hardware simulators. These findings demonstrate QAOA- integrated QRL as a viable approach to scalable, high quality quantum-assisted combinatorial optimization.

T. Satyanarayana Murthy, B. Swathi Sowmya, Santhosh Voruganti +3

Chaitanya Bharathi Institute of Technology Hyderabad, India

Quantum Machine Learning Combinatorial Optimization Hybrid Quantum-Classical ML Vehicle Routing Quantum Approximate Optimization Algorithm Quantum RL

Apr 30, 2026cs.DS

Matroid Algorithms Under Size-Sensitive Independence Oracles

The standard oracle model for matroid algorithms assumes that each independence query can be answered in constant time, regardless of the size of the queried set. While this abstraction has underpinned much of the theoretical progress in matroid optimization, it masks the true computational effort required by these algorithms. In particular, for natural and widely studied classes such as graphic matroids, even a single independence query can require work linear in the size of the set, making the constant-time assumption implausible. We address this gap by introducing a size-sensitive cost model where the cost of a query

Q

scales with

|Q|

. Nearly linear-time oracle implementations exist for broad families of matroids, and this refined abstraction therefore captures the true cost of query evaluation while allowing for a more faithful comparison between general matroids and their natural special cases. Within this framework we study three fundamental algorithmic tasks: finding a basis of a matroid, approximating its rank, and approximating its partition size. We establish tight results, proving nearly matching upper and lower bounds that show the optimal query cost is (up to logarithmic factors) quadratic in the size of the matroid. On the algorithmic side, our upper bounds are realized by explicit procedures that construct the desired solution. On the complexity side, our lower bounds are unconditional and already hold even for weaker distinguishing formulations of the problems. Finally, for matroids with maximum circuit size at most

c

, we show that the quadratic barrier can be broken, providing an algorithm that calculates the maximum-weight basis with expected query cost

\mathcal{O}(n^{2-1/c} \log n)

.

Kiarash Banihashem, MohammadTaghi Hajiaghayi, Mahdi JafariRaviz +1

*Equal contribution 1Department of Computer Science, University of Maryland, College Park, USA.

Query Complexity Oracle Complexity Combinatorial Optimization Approximation Algorithms

Apr 30, 2026cs.CE

Design Structure Matrix Modularization with Large Language Models

Design Structure Matrix (DSM) modularization, the task of partitioning system elements into cohesive modules, is a fundamental combinatorial challenge in engineering design. Traditional methods treat modularization as a pure graph optimization, without access to the engineering context embedded in the system. Building on prior work on LLM-based combinatorial optimization for DSM sequencing, this paper extends the method to modularization across five cases and three backbone LLMs. Our method achieves near-reference quality within 30 iterations without requiring specialized optimization code. Counterintuitively, domain knowledge, beneficial in sequencing, consistently impairs performance on more complex DSMs. We attribute this to semantic misalignment between the LLM's functional priors and the purely structural optimization objective, and propose the semantic-alignment hypothesis as a testable condition governing knowledge effectiveness with LLMs. Ablation studies identify the most effective input representation, objective formulation, and solution pool design for practical deployment. These findings offer practical guidance for deploying LLMs in engineering design optimization.

Shuo Jiang, Jianxi Luo

Department of Systems Engineering City University of Hong Kong, Hong Kong

Large Language Model-Guided Optimization Combinatorial Optimization

Apr 28, 2026cs.LG

Online combinatorial optimization with stochastic decision sets and adversarial losses

Most work on sequential learning assumes a fixed set of actions that are available all the time. However, in practice, actions can consist of picking subsets of readings from sensors that may break from time to time, road segments that can be blocked or goods that are out of stock. In this paper we study learning algorithms that are able to deal with stochastic availability of such unreliable composite actions. We propose and analyze algorithms based on the Follow-The-Perturbed-Leader prediction method for several learning settings differing in the feedback provided to the learner. Our algorithms rely on a novel loss estimation technique that we call Counting Asleep Times. We deliver regret bounds for our algorithms for the previously studied full information and (semi-)bandit settings, as well as a natural middle point between the two that we call the restricted information setting. A special consequence of our results is a significant improvement of the best known performance guarantees achieved by an efficient algorithm for the sleeping bandit problem with stochastic availability. Finally, we evaluate our algorithms empirically and show their improvement over the known approaches.

Gergely Neu, Michal Valko

SequeL team, INRIA Lille – Nord Europe, France

Multi-Armed Bandits Exploration-Exploitation Tradeoff Combinatorial Optimization

Apr 27, 2026cs.AI

An Analysis of the Coordination Gap between Joint and Modular Learning for Job Shop Scheduling with Transportation Resources

Efficient job-shop scheduling with transportation resources is critical for high-performance manufacturing. With the rise of "decentralized factories", multi-agent reinforcement learning has emerged as a promising approach for the combined scheduling of production and transportation tasks. Prior work has largely focused on developing novel cooperative architectures while overlooking the question of when joint training is necessary. Joint training denotes the simultaneous training of job and automatic guided vehicle scheduling agents, whereas modular training involves independently training each agent followed by post-hoc integration. In this study, we systematically investigate the conditions under which joint training is essential for optimal performance in the job-shop scheduling problem with transportation resources. Through a rigorous sensitivity analysis of resource scarcity and temporal dominance, we quantify the coordination gap -- the performance difference between these two training modalities. In our evaluation, joint training outperforms the majority of dispatching rule combinations and modular training approaches. However, the coordination gap advantage diminishes in bottleneck environments, particularly under severe transport and processing constraints. These findings indicate that modular training represents a viable alternative in environments where a single scheduling task dominates. Overall, our work provides practical guidance for selecting between training modalities based on environmental conditions, enabling decision-makers to optimize reinforcement learning-based scheduling performance.

Moritz Link, Jonathan Hoss, Noah Klarmann

Faculty of Management and Engineering, University of Applied Sciences, 83024 Rosenheim, Germany

Multi-Agent Coordination Combinatorial Optimization Job Shop Scheduling Multi-Agent Reinforcement Learning

Apr 27, 2026cs.AI

A2DEPT: Large Language Model-Driven Automated Algorithm Design via Evolutionary Program Trees

Designing heuristics for combinatorial optimization problems (COPs) is a fundamental yet challenging task that traditionally requires extensive domain expertise. Recently, Large Language Model (LLM)-based Automated Heuristic Design (AHD) has shown promise in autonomously generating heuristic components with minimal human intervention. However, most existing LLM-based AHD methods enforce fixed algorithmic templates to ensure executability, which confines the search to component-level tuning and limits system-level algorithmic expressiveness. To enable open-ended solver synthesis beyond rigid templates, we propose Automated Algorithm Design via Evolutionary Program Trees (A2DEPT), which treats LLMs as system-level algorithm architects. A2DEPT explores the vast program space via a tree-structured evolutionary search with hybrid selection and hierarchical operators, enabling iterative refinement of complete algorithms. To make open-ended generation practical, we enforce executability with a lightweight program-maintenance loop that performs feedback-driven repair. In experiments, A2DEPT consistently outperforms representative LLM-based baselines on both standard and highly constrained benchmarks. On the standard benchmarks, it reduces the mean normalized optimality gap by 9.8% relative to the strongest competing AHD baseline.

Bin Chen, Shouliang Zhu, Beidan Liu +4

The Institute of Intelligent Computing, University of Electronic Science and Technology of China, Chengdu 611731, China · College of Systems Engineering, National University of Defense Technology, Changsha 410073, China · Academy of Military Medical Sciences, Beijing 100071, China

Automated Algorithm Discovery Automated Heuristic Design Metaheuristic Optimization Large Language Model-Guided Optimization Genetic Programming LLM-Based Program Synthesis+1

Apr 27, 2026cs.LG

Query-Efficient Quantum Approximate Optimization via Graph-Conditioned Trust Regions

In low-depth implementations of the Quantum Approximate Optimization Algorithm (QAOA), the dominant cost is often the number of objective evaluations rather than circuit depth. We introduce a graph-conditioned trust-region method for reducing this query cost. A graph neural network predicts a Gaussian distribution N(mu, Sigma) over QAOA angles. The mean initializes a local optimizer, the covariance defines an ellipsoidal trust region that constrains the search, and the predicted uncertainty determines an instance-dependent evaluation budget. Thus the learned distribution defines a search policy rather than only an initial parameter estimate. Under explicit assumptions on local smoothness, curvature, calibration, and noise, we derive bounds on objective degradation within the trust region, lower bounds on gradient variance, preservation of expected objective ordering under depolarizing noise, and finite-sample coverage guarantees. We evaluate the method for MaxCut at depth p = 2 on Erdos-Renyi, 3-regular, Barabasi-Albert, and Watts-Strogatz graphs with n = 8-16 vertices. Relative to random restarts and the strongest learned point-prediction baseline, the method reduces the mean number of circuit evaluations from 343 and 85 to 45 +/- 7, while maintaining sampled approximation ratios within 3 percentage points of concentration-based heuristics. The method does not improve absolute approximation ratios; its advantage is reduced query cost at comparable solution quality. The predictive uncertainty is calibrated in the experiments, with ECE = 0.052 and Spearman correlation rho = 0.770, and the learned trust regions transfer to graph sizes not used during training. The results identify a low-depth, query-dominated regime in which graph-conditioned trust regions reduce the query cost of QAOA without modifying the ansatz.

Molena Huynh

North Carolina State University

Black-Box Optimization Combinatorial Optimization Trust-Region Optimization Quantum Approximate Optimization Algorithm

Apr 27, 2026cs.LG

Crystal structure prediction using graph neural combinatorial optimization

Crystalline materials are widely used in technological applications, yet their discovery remains a significant challenge. As their properties are driven by structure, crystal structure prediction (CSP) methods play a central role in computational approaches aiming to accelerate this process. Previously, CSP has been approached from a combinatorial optimization perspective, with the core challenge of allocating atoms on a fine grid of predefined discrete positions within a unit cell while minimizing their interaction energy. Exact mathematical optimization methods provide guaranteed solutions, but they become computationally expensive for large-scale instances, where the atomic configuration space grows rapidly, particularly in the absence of additional symmetry constraints. In this work, we introduce a neural combinatorial optimization approach to the atom allocation challenge and, subsequently, CSP, based on graph neural networks (GNNs), which can effectively sample from the distribution of feasible structures in an unsupervised manner. We leverage expander graphs to construct computational graphs over discrete positions that capture both short- and long-range interactions between atoms, and employ the Gumbel-Sinkhorn approach to enforce the desired stoichiometry of the generated structures. We demonstrate that our method outperforms classical heuristic approaches and is competitive with a commercial optimization solver across a range of chemical compositions. This enables the use of ever-expanding GPU infrastructure to tackle the inherent combinatorial challenges of CSP, paving the way for scaling beyond current capabilities.

Stavros Gerolymatos, J. Kyle Brubaker, Martin J. A. Schuetz +1

School of Computer Science and Informatics, University of Liverpool, Liverpool, England, UK · Amazon Advanced Solutions Lab, Seattle, Washington 98170, USA · Arm Inc., San Jose, California 95134

Graph Neural Networks Crystal Structure Prediction Combinatorial Optimization Crystal Structure Generation Neural Combinatorial Optimization

Apr 24, 2026cs.LG

ML-Guided Primal Heuristics for Mixed Binary Quadratic Programs

Mixed Binary Quadratic Programs (MBQPs) are an important and complex set of problems in combinatorial optimization. As solving large-scale combinatorial optimization problems is challenging, primal heuristics have been developed to quickly identify high-quality solutions within a short amount of time. Recently, a growing body of research has also used machine learning to accelerate solution methods for challenging combinatorial optimization problems. Despite the increasing popularity of these ML-guided methods, a large body of work has focused on Mixed-Integer Linear Programs (MILPs). MBQPs are challenging to solve due to the combinatorial complexity coupled with nonlinearities. This work proposes ML-guided primal heuristics for Mixed Binary Quadratic Programs (MBQPs) by adapting and extending existing work on ML-guided MILP solution prediction to MBQPs. We introduce a new neural network architecture for MBQP solution prediction and a new training data collection procedure. Moreover, we extend existing loss functions in solution prediction and propose to combine contrastive and weighted cross-entropy losses. We evaluate the methods on standard and real-world MBQP benchmarks and show that the developed ML-guided methods significantly outperform existing primal heuristics and state-of-the-art solvers. Furthermore, models trained with our proposed extension with combined losses outperform other ML-based methods adapted from MILPs and improve generalization in cross-regional inference on a real-world wind farm layout optimization problem.

Weimin Huang, Natalie M. Isenberg, Ján Drgoňa +2

University of Southern California, USA · Pacific Northwest National Laboratory, USA · Johns Hopkins University, USA

Combinatorial Optimization Mixed-Integer QP Neural Combinatorial Optimization

Apr 24, 2026cs.AI

Fast and Effective Redistricting Optimization via Composite-Move Tabu Search

Spatial redistricting is a practical combinatorial optimization problem that demands high-quality solutions, rapid turnaround, and flexibility to accommodate multi-criteria objectives and interactive refinement. A central challenge is the contiguity constraint: enforcing contiguity in integer-programming or heuristic search can severely shrink the feasible neighborhood, weaken exploration, and trap the search in poor local optima. We introduce a composite-move Tabu search (CM-Tabu) that systematically expands the feasible neighborhood space in Tabu search while preserving contiguity. When a boundary unit cannot be reassigned individually without disconnecting its district, our method identifies a minimal set of units that can move together, or a pair of units (or sets of units) that can be switched, as a contiguity-preserving composite move. Candidate single-unit and composite moves are generated in linear time by analyzing each district's contiguity graph using articulation points and biconnected components. Extensive experiments demonstrate that the proposed approach substantially improves solution quality, run-to-run robustness, and computational efficiency relative to traditional Tabu search and other baselines. For example, in the Philadelphia case, the approach can consistently attain the theoretical global optimum in population-equality and support multi-criteria trade-offs. CM-Tabu delivers optimization performance suitable for real-world practices and decision-support workflows.

Hai Jin, Diansheng Guo

Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China

Combinatorial Optimization Constrained Optimization

Apr 23, 2026cs.LO

A general optimization solver based on OP-to-MaxSAT reduction

Optimization problems are fundamental in diverse fields, such as engineering, economics, and scientific computing. However, current algorithms are mostly designed for specific problem types and exhibit limited generality in solving multiple types of optimization problems. To enhance generality, we propose an automated reduction method named OP-to-MaxSAT reduction and a general optimization solver based on OP-to-MaxSAT reduction (GORED). GORED unifies the solving of multiple types of optimization problems by reducing the problems from optimization problems to MaxSAT instances in polynomial time and solving them using the state-of-the-art MaxSAT solver. The generality and solution quality of GORED are validated through experiments on 136 instances across 11 types of optimization problems. Experimental results demonstrate that GORED not only successfully solves a wide range of optimization problems but also yields solutions comparable in quality to those from existing methods, with no statistically significant differences observed. By introducing automated reduction, this work shifts the paradigm of optimization solvers from designing specialized algorithms for each problem type to employing a single algorithm for diverse problems. As a result, advances in this single algorithm can now drive progress in a wide range of optimization problems across various domains.

Yuxin Zhao, Han Huang, Zhifeng Hao

School of Software Engineering, South China University of Technology, Guangzhou, 510006, China. · School of Software Engineering, Sun Yat-sen University, Zhuhai, 519082, China. · Department of Mathematics, College of Science, Shantou University, Shantou, 515063, China.

Combinatorial Optimization Maximum Satisfiability

Apr 23, 2026cs.AI

The CriticalSet problem: Identifying Critical Contributors in Bipartite Dependency Networks

Identifying critical nodes in complex networks is a fundamental task in graph mining. Yet, methods addressing an all-or-nothing coverage mechanics in a bipartite dependency network, a graph with two types of nodes where edges represent dependency relationships across the two groups only, remain largely unexplored. We formalize the CriticalSet problem: given an arbitrary bipartite graph modeling dependencies of items on contributors, identify the set of k contributors whose removal isolates the largest number of items. We prove that this problem is NP-hard and requires maximizing a supermodular set function, for which standard forward greedy algorithms provide no approximation guarantees. Consequently, we model CriticalSet as a coalitional game, deriving a closed-form centrality, ShapleyCov, based on the Shapley value. This measure can be interpreted as the expected number of items isolated by a contributor's departure. Leveraging these insights, we propose MinCov, a linear-time iterative peeling algorithm that explicitly accounts for connection redundancy, prioritizing contributors who uniquely support many items. Extensive experiments on synthetic and large-scale real datasets, including a Wikipedia graph with over 250 million edges, reveal that MinCov and ShapleyCov significantly outperform traditional baselines. Notably, MinCov achieves near-optimal performance, within 0.02 AUC of a Stochastic Hill Climbing metaheuristic, while remaining several orders of magnitude faster.

Sebastiano A. Piccolo, Andrea Tagarelli

University of Calabria, DeMaCS – Dept. Mathematics and Computer Science · University of Calabria, DIMES – Dept. Computer Engineering, Modeling, Electronics, and Systems Engineering

Combinatorial Optimization