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

Oct 8, 2026quant-ph

Beyond QAOA: A Review of AI and Quantum Computing for Adaptive Combinatorial Optimization

Near-term quantum approaches to combinatorial optimization are limited by qubit counts, circuit fidelity, sampling cost, and the difficulty of encoding constraints, while machine learning is increasingly used to configure and control quantum optimization workflows. We call such workflows adaptive: decisions conventionally fixed in advance, from formulation and penalties to shot budgets, backends, and whether to invoke a quantum processor at all, are made by learned policies that respond to the instance, the progress of the solve, or the hardware. This review examines three paradigms, AI for quantum optimization, quantum for AI-driven optimization, and AI-quantum co-optimization, and organizes the literature by the decision being learned rather than by application. A structured review of 119 papers, 67 coded in detail, shows that the evidence is considerably stronger for AI-assisted quantum optimization than for the reverse direction: learning already reduces quantum evaluations, improves initialization, supports decomposition and penalty control, and mitigates noise, whereas evidence that quantum computation improves learned optimizers remains largely confined to small-scale simulation. Experimental controls are thin: 25 of 57 studies include no classical baseline, the quantum contribution is fully isolated in 10 of 24 studies where an ablation applies, and the median experiment uses 17 qubits. We introduce an M0-M5 evidence hierarchy, from simulation to matched-resource practical advantage, and find no broadly convincing result at the highest level. We argue that scaling is increasingly a systems problem: the question is not only whether a problem fits on a quantum processor, but how classical and quantum resources should be allocated across the optimization process. The review is aimed at researchers in quantum computing, machine learning, and operations research.

Hoong Chuin LAU

School of Computing and Information Systems, Singapore Management University, Singapore

Combinatorial Optimization Hybrid Quantum-Classical ML Quantum Machine Learning

Oct 7, 2026cs.CV

Rubix: Global Correspondence-Free Point Set Alignment through Assignment Geometry

Procrustes-Wasserstein alignment jointly estimates a matching and rotation without supplied correspondences, but alternating minimization can stop at suboptimal solutions. Rubix solves the equally weighted planar problem globally under squared Euclidean loss. Each matching

σ

of two centered

n

-point sets defines a complex correlation

z_σ=\sum_i\bar x_i y_{σ(i)}

. Their convex hull is the permutation polygon: supporting vertices give optimal matchings at fixed rotations, and the farthest vertex gives the global alignment. We prove the sharp bound of

n(n-1)

vertices for

n\ge2

, answering Rote's rotation-assignment open problem. In exact arithmetic, assignment queries recover the polygon in

\mathcal O(n^5)

operations. Assignment-based bounds extend the approach to three-dimensional rotations and partial matching at a supplied translation through branch-and-bound. On timed MPEG-7 shape pairs, Rubix attains every numerical reference value in 12 ms on average, 50 times faster than a rotation grid at the same accuracy. Its distances improve gravity-aligned matching of real 3D scans, shape retrieval and noisy crystal classification over alternating minimization.

Subhransu S. Bhattacharjee, Dylan Campbell, Rahul Shome

School of Computing, Australian National University

Point Cloud Registration Combinatorial Optimization

Oct 7, 2026cs.AI

Agentic AI-Assisted Modeling for Production Scheduling: Assessment in Constraint Programming

Developing optimization models for production scheduling requires substantial expert effort. Research on large language models (LLMs) has followed two directions: specialized approaches for automated modeling, mostly for mixed-integer linear programming, which often rely on dedicated training or problem-specific architectures that limit industrial deployment; and agentic artificial intelligence for operational decision support, which generally assumes that the optimization model already exists. This study bridges both directions by assessing whether general-purpose LLMs, orchestrated as agents without task-specific training, can formulate and implement constraint programming models from natural-language problem descriptions. Singleagent and multi-agent architectures are integrated with a Model Context Protocol server that provides context-aware retrieval of solver documentation to mitigate hallucinations during implementation. Both are compared with a direct LLM baseline on six industry-oriented problems covering flow-shop, job-shop, flexible job-shop and resource-constrained warehouse scheduling, using three LLMs and assessing modeling accuracy, execution success, latency and token consumption. Formulation proves largely within reach of current LLMs, whereas implementation is the main barrier. The multi-agent workflow raises the share of scripts that run correctly as generated from 14.8% with a direct LLM call to 59.3%, reaching 80.6% on the four less complex problems, while tightly coupled intralogistics models remain an open challenge.

Ángel Sánchez-Fernández, Javier Pernas-Álvarez, Diego Crespo-Pereira

Universidade da Coru˜na, Campus Industrial de Ferrol, CITENI, Grupo Integrado de Ingenier´ıa, Esteiro s/n, Ferrol, 15403.

Natural Language to Optimization Modeling Combinatorial Optimization Constraint Programming Multi-Agent LLM Systems LLM Agent Orchestration

Oct 7, 2026cs.MA

CANDO: Cooperative Agentic Network for Layout Design Optimization

Layout generation for real-world facilities is a challenging problem, requiring reasoning over irregular site boundaries, heterogeneous orientations, access-aware placements, and motion-planning feasibility. Yet, most existing layout benchmarks in the generative AI space target simpler placements over rectangular domains and rely on distributional metrics such as FID and IoU that reward conformity to dataset priors, thus discounting design innovation. Motivated by these gaps, we introduce ALPS-Bench, a benchmark of

1,000

professionally annotated real-world facility layouts paired with an instance-specific scoring protocol grounded in a structured design manual. As a strong baseline for ALPS-Bench, we propose CANDO, a training-free multi-agent framework in which specialized agents iteratively refine layouts through a verification-grounded loop, concentrating reasoning on strategic spatial decisions. We demonstrate that CANDO surpasses state-of-the-art trained and LLM-based baselines on the widely adopted PubLayNet, RICO, and PKU-PosterLayout benchmarks, establishing cooperative agentic design as a broadly effective recipe for constraint-aware layout synthesis.

Athanasios Masouris, Zheng Jing, Benjamin Sam Chandler +1

Shell Information Technology International · Shell China Limited · Delft University of Technology (TU Delft)

Benchmark Design Combinatorial Optimization

Oct 6, 2026cs.CC

On the complexity of the single-move labeled token routing problem

In neutral-atom quantum computers, atoms are moved to target positions along paths of empty positions, and a target position may be reserved for one species of atom. Motivated by this task, we introduce Single-Move Labeled Token Routing: every source and every target vertex of a graph is assigned a set of labels, and tokens occupy the sources. A solution consists of a matching that assigns each source to a compatible target (one whose label set intersects its own), a route for each matched pair, and a movement order in which, when a token is moved, its route contains no other token. The problem is known to be polynomial-time solvable when every source is compatible with every target, and

\mathsf{NP}

-complete on grid graphs when each source is compatible with exactly one target. We prove that the latter case remains

\mathsf{NP}

-complete on grids and on planar graphs of maximum degree four even when some solution has pairwise edge-disjoint routes. On trees, the problem is known to be

\mathsf{NP}

-complete even for maximum degree three. We study trees through the solution edge multiplicity, the largest number of routes of a solution sharing an edge, and the candidate edge multiplicity, the largest number of compatible pairs whose paths share an edge. We prove that on trees of maximum degree three, the problem is

\mathsf{W}[1]

-hard parameterized by a bound on the solution edge multiplicity, even when a movement order is given, and that on trees of unbounded degree, it is

\mathsf{NP}

-complete even when the candidate edge multiplicity is at most eight. We show that on trees the problem is fixed-parameter tractable parameterized by the maximum degree together with the candidate edge multiplicity, and also by the candidate vertex multiplicity, the same count at vertices. Unless

\mathsf{P}=\mathsf{NP}

, neither the maximum degree nor the candidate edge multiplicity can be omitted.

Nicolas Bousquet, Remy El Sabeh, Amer E. Mouawad +1

American University of Beirut, Beirut, Lebanon

Combinatorial Optimization Parameterized Complexity

Oct 6, 2026cs.LG

A Decision-Focused Neural Optimization Framework for Personalized Route Reproduction from Vehicle Trajectories

This study formulates individual route reproduction as a shortest-path problem over learned driver-specific latent link costs. The central idea is that, once such latent costs are inferred from contextual information, observed routes can be reproduced without enumerating alternative route sets. We propose a neural pipeline that includes a perception model that embeds context covariates, which comprises individual characteristics, trip-specific attributes, and network-level traffic states, into the personalized link costs. A constrained optimization (CO) layer, which determines the shortest path (SP) based on these estimated costs, follows the perception encoder. To enable end-to-end training, we employ decision-focused learning to align the predicted shortest paths with observed routes. The implicit maximum likelihood estimation (iMLE) provides an approximate gradient of the loss function that contains the non-differentiable CO layer. Furthermore, a regularization term anchors the latent cost distribution to the empirical scale of observed link travel times, mitigating the scale ambiguity inherent in shortest-path supervision. Empirical evaluations demonstrate that the proposed framework outperforms baseline route choice models in path reproduction. The learned latent costs, interpreted as proxies for perceived travel costs, provide plausible explanations for heterogeneous route choices.

Gyeongjun Kim, Yeseul Kang, Keemin Sohn

Department of Smart City, Chung-Ang University, Seoul, Korea · Department of Urban Engineering, Chung-Ang University, Seoul, Korea · Department of Urban Engineering, Department of Smart City, Chung-Ang University, Seoul, Korea

Learning to Optimize Combinatorial Optimization

Oct 5, 2026cs.RO

Towards Decentralized Formation of Minimum-Length Communication Networks Using Robot Swarms

Multi-robot missions in infrastructure-denied environments frequently rely on reliable communication links between spatially separated locations. We propose a fully decentralized framework for constructing and dynamically maintaining communication networks without centralized topology planning or global positioning infrastructure. Driven strictly by local interactions, robots reconfigure local network topologies and adjust their physical positions to minimize overall network length while adhering to communication constraints. Formal analysis shows that our local reconfiguration operations guarantee continuous network connectivity, strictly decrease network length with every branch transfer, and bound worst-case performance to the shortest starlike tree. Embodied simulations and physical multi-robot experiments confirm that our approach forms networks near the length of centrally computed Euclidean Steiner trees. Additionally, the system dynamically adapts to moving targets and optimizes deployment by utilizing only necessary connectors, preserving excess robots for auxiliary tasks. This work enables autonomous swarms to self-organize adaptive ad hoc communication infrastructure in communication-denied environments, which could support applications varying from subterranean exploration to planetary missions.

Genki Miyauchi, Mohamed S. Talamali, Julian Rau +1

Bristol Robotics Laboratory, University of Bristol, Bristol, UK · School of Electrical and Electronic Engineering, University of Sheffield, Sheffield, UK · Department of Computer Science, Technical University of Darmstadt, Darmstadt, Germany

Swarm Robotics Combinatorial Optimization Multi-Robot Coordination Decentralized Optimization

Oct 2, 2026cs.AI

Multi-Task Evolution for Zero-Shot Cross-Problem Generalization using LLMs

Designing effective heuristics for diverse combinatorial optimization problems requires substantial expertise and repeated search. Large language models (LLMs) automate heuristic generation and refinement, but heuristic search typically depends on evaluation feedback from the problem being optimized. Generalizing to new problem definitions using only source-task feedback therefore remains a central challenge. We introduce MECo, an LLM-driven multi-task evolutionary framework for zero-shot cross-problem generalization. MECo maintains task-conditioned heuristic populations and uses a transfer gap based on cross-task population performance to guide their interactions. These interactions enable the transfer and recombination of heuristics. A complementary selection criterion then constructs a compact heuristic set by rewarding each member's additional coverage of source combinations. The selected set is applied to target problems without further search or adaptation. Experiments on 32 problem variants across vehicle routing (VRP) and flexible job-shop scheduling (FJSP) show that MECo achieves the lowest mean costs compared with eight automated heuristic design (AHD) baselines under the same budgets. On out-of-domain problems, it outperforms the strongest baseline in each family. Moreover, integrating the framework of MECo with different AHD methods improves their ID and OOD performance in both families, supporting its effectiveness across different methods.

Zhuoliang Xie, Changliang Zhou, Genghui Li +1

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 · College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, China

Automated Heuristic Design Large Language Model-Guided Optimization Combinatorial Optimization Multi-Task Optimization Evolutionary Optimization

Oct 1, 2026cs.DS

When Is Deletion Ordering Tractable? From Update Dynamics to Permutation Structure

Given a fixed set of pending deletion requests, retraining from scratch after each request is prohibitive, so a prescribed request-wise policy processes them sequentially. The resulting terminal model can depend on their order. Rather than prescribing an ordering rule, we study the permutation objective induced by the fixed policy and ask when it admits simpler structure. We identify two independent reductions: position additivity represents the objective by request--position costs, reducing optimization to assignment and, with a shared positional profile, sorting; suffix localization removes dependence on the distant prefix while retaining interactions among the surviving requests. Under shared affine updates, we characterize the quadratic interactions that obstruct additivity, prove the reductions' independence, and show that suffix-conditioned assignment improves the approximation rate from O(p^L) toO(p^(2L)). Experiments recover both structures in executed objectives. A controlled damped-Newton sweep shows that stronger contraction shifts the objective toward shorter, more suffix-specific dependence, while two full-network policies exhibit distinct positional and within-suffix structure. Structures identified from compact execution sets also predict unseen orders. These results frame deletion ordering as identifying the computational structure induced by the executed updates.

Xinyu Wang, Ziyu Zhao, Yixuan He +1

McGill University · Arizona State University

Combinatorial Optimization Machine Unlearning

Sep 30, 2026cs.LG

Validity-Preserving Hierarchical RL for Joint Routing and Switch Placement in EDA

Routing and switch placement are fundamental combinatorial optimization problems in chip design, requiring the joint optimization of routing topology and physical placement under strict structural, geometric and logical constraints. Existing approaches typically rely on carefully engineered heuristics that incorporate strong problem-specific biases to navigate the enormous space of possible designs. In this work, we introduce a hierarchical reinforcement learning framework for joint routing and switch placement at the level of logical communication routes. Starting from a minimal routing graph, our method progressively constructs increasingly expressive solutions through three coupled operations: switch expansion, switch placement, and route refinement. These operations preserve routing validity by construction, restricting exploration to feasible configurations where every communicating initiator-target pair has one assigned loop-free route. We explore the induced solution space using Gumbel Monte Carlo Tree Search, showing that neural-guided search substantially improves solution quality over non-learning optimization methods. Furthermore, pretraining across floorplans provides a strong initialization for fine-tuning on unseen instances.

Dorian Gailhard, Ugo Lecerf, Enzo Tartaglione +2

LTCI, Télécom Paris, Institut Polytechnique de Paris, France · Arteris IP

Electronic Design Automation Hierarchical RL Combinatorial Optimization RL for Combinatorial Optimization

Sep 30, 2026cs.NE

An Island-Based Parallel Biased Random-Key Genetic Algorithm for the Three-Dimensional Trailer Loading Problem

The Three-Dimensional Trailer Loading Problem (3D-TLP) involves determining the optimal placement and orientation of heterogeneous items within the confined space of a trailer while maximizing volume utilization and satisfying a wide range of complex logistical and safety constraints. The 3D-TLP is NP-hard, rendering exact optimization approaches computationally impractical for large-scale industrial applications. To address this challenge, we propose an enhanced Biased Random-Key Genetic Algorithm (BRKGA) accelerated through a novel island-based parallelization framework, PANGEA. The proposed method combines the search efficiency and robustness of BRKGA with a multi-population evolutionary scheme for genetic algorithms. This island-model strategy promotes population diversity, mitigates premature convergence, and significantly reduces computational times. The proposed solution was validated in a real trailer loading process, providing an effective solution approach for real-world large-scale logistics.

A. del Río, L. Díaz, L. C. de Vicente +2

Intelligent Systems Department Gradiant Vigo, Spain

Evolutionary Optimization Combinatorial Optimization Genetic Algorithms

Sep 30, 2026cs.RO

CLIPPER Beyond Shortlisting: Auditable Decision Support for Changing Municipal Micromobility Policies

In municipal planning workshops, planners and other stakeholders compare shared-micromobility parking policies by varying no-parking zones, retained sites, spacing, or area allocations. Each edit changes feasible sites and how much demand they cover, so the alternative must be reoptimized on the same spatial data. Full-set greedy, the transparent reference for this task, takes tens of seconds per alternative at city scale. We present Constraint-exact Low-latency Iterative Planning with Pooled Evaluation and Replay (CLIPPER), an optimizer with audit functions developed for requirements elicited with the City of Braunschweig. In each greedy round, it forms a deterministic candidate pool of bounded size, computes how much still-uncovered demand each candidate would add, and rejects candidates that violate an active constraint. An optional offline audit scans every remaining feasible candidate and records what the restricted pool omitted. We evaluate these functions on complete eleven-state edit chains (

E_0,\ldots,E_{10}

) in Braunschweig, Munich, and Berlin. With

K=1024

candidates per group, the fixed-width mode CLIPPER-F has mean coverage gaps to full-set greedy under the same policy of 0.245, 0.003, and 0.001 percentage points in Braunschweig, Munich, and Berlin, respectively, while mean rollout time falls by factors of 13.6--28.9; no audited run terminates while a candidate outside the pool could still increase coverage. Plans computed from two checksummed versions of Braunschweig's official no-parking-zone data differ in 30 of about 540 selected sites although coverage moves by only about 0.1 percentage points. These changes still require municipal assessment and implementation. The findings inform a proposed municipal process that versions policy inputs, reports site changes beside coverage, and scans the full candidate set before a final decision.

Julian Teusch, Jörg Philipp Müller, Monika Sester

Institute of Computer Science Clausthal University of Technology Clausthal-Zellerfeld, Germany · Institute of Cartography and Geoinformatics Leibniz University Hannover Hannover, Germany

Combinatorial Optimization Urban Planning

Sep 30, 2026cs.LG

GeoNest: Learning to Select Failure-Aware Neighborhoods for the Irregular Knapsack Problem in a Circular Container

The two-dimensional irregular knapsack problem in a fixed circular container is an important combinatorial optimization problem for maximizing material utilization in manufacturing. Conventional geometric packing solvers can produce tightly packed layouts, yet they often partition the residual space into isolated small pockets that cannot fit valuable unplaced polygons. To overcome this late-stage packing bottleneck, we propose a failure-aware large neighborhood search framework named GeoNest, driven by a graph policy trained via reinforcement learning. Specifically, we first construct neighborhoods by pairing failed target polygons with residual pockets. We then use explanatory poses to identify the placed polygons that block candidate insertions. These diagnosed blocking relations define bounded, fixed-item repair subproblems for the underlying geometric solver. Finally, the graph policy selects the most promising subproblem for execution. For evaluation, we introduce CircleNest-Bench, a benchmark comprising 2,391 load-controlled instances from four contour sources, including a held-out industrial CAD source. Experimental results demonstrate that, under the same total time budget, GeoNest improves mean utilization over a state-of-the-art standalone packing solver by about 0.9% on average across the three main test sets and by about 0.6% on the held-out industrial set.

Zhongman Du, Huiming Zhang, Linlin Yang +2

Beihang University, Beijing, China · Communication University of China, Beijing, China · Hangzhou Innovation Institute of Beihang University, Hangzhou, China

Metaheuristic Optimization Combinatorial Optimization RL for Combinatorial Optimization Large Neighborhood Search

Sep 30, 2026cs.LG

Blackboard Intelligence Can Surpass Autoregressive on Globally Constrained Problems

Next-token prediction has driven remarkable progress in large language models, yet a growing body of evidence suggests that they can struggle on problems governed by complex global constraints. In this work, we focus on this regime and ask whether some of these limitations arise from the inference interface induced by next-token prediction itself. We study this question through blackboard intelligence: an inference-time perspective in which a model works on a fixed, revisable canvas and searches over candidate solution states rather than committing to a causal, left-to-right trajectory. We instantiate this idea with diffusion language models, whose any-order prediction interface naturally exposes predictions over partially filled solution states. Our key observation is that mean confidence, a simple model-internal quantity available from the standard masked diffusion objective, provides a useful proxy for global coherence and can guide inference-time search and revision. Empirically, across ZebraLogic, Nurse Rostering, and Job-Shop Scheduling, Blackboard consistently improves inference while holding the fine-tuned LLaDA-8B-Instruct checkpoint fixed and substantially outperforms same-scale autoregressive baselines, reaching 90.4% accuracy on ZebraLogic-Hard, 76.4% exact feasibility on Nurse Rostering, and 80.2% optimality on JSSP. Stronger autoregressive search and refinement also fail to close the gap on ZebraLogic-Hard, while Blackboard surpasses tested frontier LLMs there and on JSSP despite their substantially greater scale and strong test-time reasoning. We open-source our codebase at https://github.com/jwoosang1/blackboard-intelligence.

Woosang Jeon, Jaeyeon Kim, Sham Kakade +6

Seoul National University · Harvard University · University of Central Florida

Inference-Time Search Combinatorial Optimization Diffusion Language Model Inference Constraint Satisfaction

Sep 29, 2026cs.AI

SimpleEvol: An Agent-Loop Framework for LLM-Driven Automated Heuristic Design with Minimal Human Priors

Large language models (LLMs) have emerged as powerful tools for automated heuristic design (AHD), enabling iterative generation and refinement of heuristics. However, the dominant paradigm embeds LLMs as narrow, fixed components, such as crossover or mutation, within heavily hand-engineered evolutionary frameworks. We argue this misapprehends LLMs. It treats them as specialized tools rather than general reasoners, constrains them to low-level operations, and underutilizes their autonomy. Moreover, the extensive human priors in these frameworks violate the bitter lesson principle that general methods scaling with computation surpass hand-crafted solutions. This raises a key question: which AHD framework designs best convert stronger LLM capabilities into better heuristics? To address this, we propose metrics for LLM-driven AHD framework handcraftedness (AHI) and intelligence conversion efficiency (ICE). Evaluating ten LLMs across three challenging combinatorial optimization problems, we obtain a notable finding that frameworks with fewer human priors consistently yield higher ICE. Based on this finding, we propose SimpleEvol, an agent-loop framework for AHD which removes nearly all human priors and allows the LLM to operate autonomously. SimpleEvol consistently achieves the highest ICE, often by a large margin. Our results challenge the trend toward complex AHD pipelines and point to a lighter and more model-centric alternative, suggesting that reducing human priors is a more effective strategy to scale up with model intelligence. The source code is available at https://github.com/HenryZhu1029/SimpleEvol-Master.

Jianghan Zhu, Cong Zhang, Rongjie Zhu +2

School of Computing and Information Systems, Singapore Management University, Singapore · Independent Researcher · School of Teacher Education, Nanjing University of Information Science and Technology, China +1

Automated Heuristic Design Large Language Model-Guided Optimization Combinatorial Optimization LLM Agents

Sep 28, 2026cs.CL

HeurEvo: Agentic Evolution of Hybrid Solver-Augmented Heuristics for Time-Critical Mathematical Optimization

Recent advances in agentic heuristic design use AI agents and execution feedback to automate algorithm discovery for challenging optimization problems. In many practical settings, high-quality solutions must be obtained under strict runtime constraints, motivating hybrid approaches that combine problem-specific heuristics with powerful mathematical programming solvers. However, existing approaches typically improve heuristic components within predefined procedures or tune solver configurations in isolation. This limits holistic adaptation of where to allocate computation, how to leverage solvers, and how to refine the overall algorithmic structure. To address these limitations, we propose HeurEvo, an automated plan--code--component co-evolution framework that jointly evolves the high-level algorithmic structures, their implementations, and a shared pool of reusable components. A planner determines which algorithmic components to use, how to combine them, and how to allocate runtime across stages, a coder realizes the resulting plan as executable code, while a component evolver updates the shared component pool. Within an island-based evolutionary framework, plans and implementations co-evolve with feedback from an interpreter agent that analyzes execution results and identifies opportunities for improvement. Across diverse combinatorial optimization benchmarks and challenging MIPLIB instances, HeurEvo finds high-quality solutions within tight runtime budgets, often matching or surpassing state-of-the-art optimization solvers given hours or days of computation. On several nonlinear geometry problems such as hexagon packing, it also improves upon the best previously reported results. These results highlight the value of jointly searching over algorithmic structure and implementation for agentic heuristic design.

Feijie Wu, Hugo Barbalho, Konstantina Mellou +5

Purdue University · Microsoft Research

Automated Algorithm Discovery Automated Heuristic Design Evolutionary Optimization Combinatorial Optimization Mixed-Integer QP

Sep 28, 2026cs.AI

Just Initialize: A Training-Free Initialization Component for Large-Scale Routing Optimization

Large-scale routing problems are difficult to solve efficiently as their search spaces grow rapidly with problem size. Existing approaches primarily improve the optimization procedure itself, often at increasing computational cost. We instead shift the focus to a useful initialization that can be refined into a high-quality solution with limited downstream refinement. We propose Just Initialize, a training-free and solver-agnostic initialization component for large-scale routing optimization. Just Initialize compresses a large routing instance into a compact surrogate space, optimizes its global routing structure, and recovers the resulting solution as an optimization-friendly starting point in the original space. Extensive experiments on Traveling Salesman Problems (TSPs), Capacitated Vehicle Routing Problems (CVRPs), Vehicle Routing Problems with Time Windows (VRPTWs), and Prize-Collecting Traveling Salesman Problems (PCTSPs) demonstrate that Just Initialize achieves high-quality solutions comparable to or better than state-of-the-art methods while substantially reducing computational cost across instances ranging from 1K to 100K nodes, including an average speedup of approximately 70

\times

, sub-second runtimes on 10K-node instances, and runtimes within tens of seconds on 100K-node instances.

Jiale Zhao, Sirui Mao, Zimu Chen +5

School of Automation Science and Electrical Engineering, Beihang University, Beijing, China · School of Computer Science and Engineering, Beihang University, Beijing, China · School of Cyber Science and Technology, Beihang University, Beijing, China +1

Traveling Salesperson Problem Combinatorial Optimization Vehicle Routing

Sep 28, 2026cs.LG

Multi-Attractor GNNs: Set-Valued Expressivity Beyond Unique Equilibria

Recurrent and equilibrium graph neural networks (GNNs) often enforce a unique fixed point or use one training target per graph. Yet many combinatorial and scientific problems admit multiple valid solutions, with no preferred one. A designated target can then impose an arbitrary selection rule. For tasks invariant to node relabeling, a symmetric graph may have a symmetric solution set but no symmetric solution. We show that multiple equilibria enable one weight-tied message-passing GNN to represent set-valued equivariant maps: different initializations approach different valid solutions. Under stated regularity assumptions, we first construct globally Lipschitz, permutation-equivariant dynamics that converge almost surely to valid solutions and reach every solution branch with positive probability. We then establish approximate realization by recurrent message passing with continuous component maps, with arbitrarily small update and limiting errors and arbitrarily high probability. This goes beyond standard universality arguments: although message passing alone cannot distinguish symmetric nodes, the evolving state keeps nodes distinguishable at every finite step without auxiliary node identifiers. Such dynamics can be learned without solution labels using problem-specific energies. On Ising ground states, structural module detection in protein graphs, and chemical reaction steady states, the learned updates produce multiple high-quality predictions with high numerical convergence rates. They achieve better average solution quality than the tested unique-equilibrium, single-target, and feedforward baselines, while remaining competitive with much larger diffusion-based solvers.

Jialin Liu

GNN Expressivity Combinatorial Optimization Equivariant GNNs

Sep 28, 2026cs.LG

Compute Time Scaling with Recursive Models for Combinatorial Optimization

We propose Tiny Recursive Models for Combinatorial Optimization (\ours{}), a general neural method for combinatorial optimization that scales both depth (how often we recursively invoke our network) and width (how much we sample in parallel). Both are fundamental for combinatorial optimization: hard instances demand a large amount of compute, while a small network is essential to avoid overfitting and capture the algorithmic essence of optimization. In particular, our method consists of a graph-aware tiny recursive model that iterates on a latent state with adaptive halting and needs only a lightweight problem-specific decoder. Compared with previous heatmap-based general neural solvers, it achieves a better balance between solution quality and inference speed on both the Traveling Salesman Problem~(TSP) and the Maximum Independent Set~(MIS) problem, and remains competitive with hybrid methods that combine neural components with heuristics specific to each problem. With the same backbone architecture for both tasks, \ours{} outperforms every diffusion-based solver on TSP from 500 to 10,000 cities at a lower inference cost, and on the standard Erdős--Rényi-[700-800] MIS benchmark it surpasses all neural solvers except those that only work well on MIS. We then explore self-relabeling for self-supervised training. We periodically replace the current set of training labels with the model's own better solutions, as an alternative training signal. Self-relabeling can, while forgoing supervision from near-optimal solutions, still result in on-par quality.

Zhengxi Zhang, Paul Swoboda

Traveling Salesperson Problem Inference-Time Scaling Combinatorial Optimization Neural Combinatorial Optimization

Sep 27, 2026cs.AI

APEX: An Extensible Model for Agent-Assisted Production Scheduling

Production scheduling requires realistic models that reflect operational constraints and efficient methods that balance competing goals. Putting these methods into use also requires data integration, model adaptation and specialist expertise. We present APEX, an extensible production scheduling framework built around a general model and hybrid multiobjective search. Agent assistance supports both scheduling and model refinement: agents prepare data and explore scenarios in natural language, while coding agents help implement and test new constraints and objectives. Shared construction and checking procedures connect these adaptations to the scheduling core. We benchmark eight APEX configurations against NSGA-II, SPEA2, MOEA/D and SMS-EMOA on 69 public job-shop, flexible job-shop and permutation flow-shop instances, assessing workload completion time (makespan), total job flowtime and computation time. Hybrid configurations achieve the best aggregate solution quality, although the leading method depends on the problem class and objective. A separate synthetic workflow study uses OpenAI's GPT-6-astra as an interaction layer between the human planner and the algorithmic core, testing rule additions, plan and objective changes, and what-if comparisons. All 24 sessions completed the requested changes and passed independent checks of saved models and schedules. A separate coding evaluation produced six native implementations of an additional objective or hard constraint through predefined extension hooks. All passed independent checks without modifying the core.

Felix J. Grumbach, Stefan Görlitz

Qunevo GmbH, Germany

Multi-Objective Evolutionary Optimization Combinatorial Optimization Job Shop Scheduling

Sep 22, 2026cs.AI

Neutral-Atom-based Quantum Optimization for Resource Allocation in NOMA Networks

In wireless communication networks, many resource optimization problems are nondeterministic polynomial-time hard (NP-hard) due to their combinatorial nature and high computational complexity. Recently, neutral-atom-based quantum computing has emerged as a promising platform for efficiently solving such problems by leveraging quantum superposition and entanglement. However, its application to wireless communication optimization problems remains largely unexplored. In this paper, we investigate the use of neutral-atom quantum platforms to solve the maximum access problem (MAP), formulated as a mixed-integer programming task that jointly considers admission control, user clustering, channel assignment, and power allocation in a non-orthogonal multiple access (NOMA)-enabled uplink network. To reduce the computational burden, the MAP is equivalently reformulated as a maximum independent set (MIS) problem in graph theory. This reformulation enables the use of the neutral atom platform based on Rydberg atom arrays, where the MIS problem is naturally encoded into the physical geometry and blockade constraints of the quantum system. Numerical results demonstrate the feasibility and potential of this approach for addressing large-scale wireless resource optimization problems.

Patatchona Keyela, Remon Polus, Soumaya Cherkaoui +1

Department of Computer and Software Engineering, Polytechnique Montréal, Montréal, Québec, H3T 1J4, Canada · Thales cortAIx Labs, Montréal, Québec, H3C 0B4, Canada

Combinatorial Optimization Wireless Communications Wireless Resource Allocation

Sep 22, 2026cs.LG

Self-Supervised Combinatorial Optimization with Constraints via Frank-Wolfe

Self-supervised learning for combinatorial optimization has emerged as a promising paradigm for solving discrete optimization problems with neural networks, but a central challenge remains: handling hard combinatorial constraints within continuous, gradient-based training. Continuously extending combinatorial objectives to convex domains is a powerful technique, yet existing approaches often require projection steps that constrain neural network outputs to lie inside the feasible polytope and rely on ad-hoc and problem-specific constructions. We propose a general framework in which the neural network is allowed to predict arbitrary continuous vectors that could potentially lie outside of the feasible polytope. These predictions are then approximated by sparse convex combinations of feasible solutions using a geometric decomposition algorithm based on Frank--Wolfe methods and approximate Caratheodory results. This decomposition induces an a.e.-differentiable, self-supervised loss defined as the expected value of the discrete objective. The same procedure provides an automatic rounding guarantee at inference time. We demonstrate strong empirical performance across multiple combinatorial problems, including the Quadratic Assignment Problem, Maximum Coverage, and the Traveling Salesperson Problem.

Akbar Rafiey, Yifei Xu, Nikolaos Karalias

NYU · MIT

Self-Supervised Learning Combinatorial Optimization Constrained Optimization Neural Combinatorial Optimization

Sep 21, 2026cs.LG

Deep Reinforcement Learning on Item-Compatibility Graphs for One-Dimensional Bin Packing

The one-dimensional bin packing problem (1D-BPP) is a classical NP-hard combinatorial optimization problem with applications ranging from logistics and manufacturing to cloud resource management. Although deep reinforcement learning (DRL) has become a competitive paradigm for data-driven optimization, most learned packing methods target 2D and 3D variants, and intelligent learned solvers for 1D-BPP remain scarce. In this paper, we present a novel end-to-end, size-agnostic graph reinforcement learning framework for 1D-BPP. We formulate the packing process as a Markov decision process on an item-compatibility graph, serving as a structural knowledge representation in which every action merges two partial bins that fit together. A graph neural network actor-critic policy extracts relational features from this representation and is trained through reinforcement learning and decoded by stochastic beam search, enabling a single trained model to generalize zero-shot to instances of any size. We conduct a systematic empirical study across graph encoders, DRL algorithms, reward functions, training distributions, and hyperparameters. Evaluated zero-shot on the full BPPLIB benchmark against a constructive heuristic, a grouping genetic algorithm, and recent learned methods, our data-driven policy lowers the mean optimality gap of the constructive heuristic from 2.66% to 2.31%, with the largest gains on structured instances. Against learned baselines evaluated on the same benchmark, it attains a lower gap on most of the nine families and is far more stable across instance distributions. On the hardest benchmark family, it outperforms a state-of-the-art learned solver that relies on column generation and integer programming, while using no solver at all. A grouping genetic algorithm remains ahead overall, and we analyze where and why the residual gap arises.

M. Aslı Aydın

Department of Management Engineering, Bahçeşehir University, Istanbul, Türkiye

Combinatorial Optimization RL for Combinatorial Optimization Generalization in RL Neural Combinatorial Optimization

Sep 21, 2026cs.DS

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that

γ

-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a

γ

-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative

(1+1/γ)^{-2}

factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to

ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }

. In sharp contrast to the previous guarantee, our obtained factor

ρ_γ

not only strictly improves upon

(1+1/γ)^{-2}

for every

γ\in(0,1]

, but also can asymptotically approach the optimal

(1-1/e)

-approximation for submodular maximization as

γ\to1

. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of

α

-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of

1-e^{-γ}

and

1-e^{-α}

, respectively. Here,

α\in(0,1]

denotes the DR ratio.

Shi Fu, Youming Qiao, Dacheng Tao +2

College of Computing and Data Science, Nanyang Technological University, Singapore · Centre for Quantum Software and Information, University of Technology Sydney, Australia · School of Computing and Information Technology, Great Bay University, China

Submodular Optimization Combinatorial Optimization Approximation Algorithms

Sep 21, 2026cs.MA

Mixed-integer flow formulations for motion planning and decision-making of networked multi-agent systems

This work investigates the use of flow-based connectivity maintenance constraints in mixed-integer linear programming (MILP) trajectory planning and decision-making models for networked multi-agent systems (MAS). We integrate flow-based encodings for standard and k-hop connectivity into MILP multi-vehicle maneuvering models that are widely used alongside receding horizon planning strategies. Their necessity and sufficiency is demonstrated, guaranteeing full coverage of potential network topologies. The flow formulation for standard connectivity decreases the growth of the required inequality constraints from exponential to polynomial w.r.t. the size of the MAS when compared to the state-of-the-art subtour elimination (SEC) method. The flow-based k-hop connectivity constraints decrease the number of required binary variables and decouple its growth from the number of hops. However, the impact of these formulations in performance is not straightforward due to the introduction of a substantial number of continuous flow optimization variables and, in the case of k-hop connectivity, additional inequality constraints. We investigate this trade-off through a statistical evaluation of costs and optimization times using a conventional branch-and-bound commercial solver and trials performed with randomized environments for increasingly larger MAS. The results show that the flow formulation outperforms SEC in standard connectivity problems, enabling the solutions to be computed for larger MAS considering the imposed optimization time limit. The reduction in number of binary variables enabled by the k-hop flow formulations decreases the theoretical worst-case number of iterations required by the branch-and-bound algorithm to compute the global optimal solution. Our results show that this advantage did not translate into improvements in the average performance when compared to the baseline.

Angelo Caregnato-Neto, Paul-Louis Delacour, Raf Van de Plas +2

Department of Computational Mechanics (DMC), School of Mechanical Engineering (FEM), State University of Campinas (UNICAMP), São Paulo, Brazil · Delft Center for Systems and Control, Delft University of Technology, Delft, Netherlands · Dept. of Biochemistry, Vanderbilt University, Nashville, TN, USA

Constrained Motion Planning Mixed-Integer Linear Programming Combinatorial Optimization Multi-Agent Systems Multi-Agent Planning

Sep 21, 2026cs.LG

Dual-GNN Multilevel Coarsening for Maximum Independent Set

The maximum independent set (MIS) problem is a fundamental NP-hard combinatorial optimization problem with applications in scheduling, resource allocation, and network analysis. Exact solvers can provide high-quality solutions or optimality certificates, but their computational cost grows rapidly with graph size, while hand-crafted heuristics improve scalability at the expense of guarantees. Learning-based methods offer an alternative by exploiting structural patterns across graph instances, yet directly predicting independent sets can make global coordination difficult on large graphs. We instead use learning to guide multilevel graph coarsening while retaining combinatorial search for final decision making. Our Dual-GNN Multilevel Coarsening framework uses a Partition GNN to score candidate contractions and a Representative GNN to select top-k local independent-set states for each final cluster. Experiments on Erdős--Rényi graphs with up to 2,000 vertices demonstrate a favorable quality--runtime trade-off. On 500-vertex instances with certified optima, our method achieves an average independent-set size of 19.20, corresponding to 99.5% of the optimal value of 19.30, while reducing the mean wall-clock time from 643.57 seconds for exact solving to 3.41 seconds, yielding an approximately 189

\times

speedup. On larger graphs with 1,000 and 2,000 vertices, our method achieves the best mean solution quality among all evaluated methods. Moreover, although trained only on Erdős--Rényi graphs with edge probability

p=0.35

, the learned coarsening policy generalizes effectively across both unseen graph densities and structurally different graph families.

Tianfeng Chen, Xianyue Li

School of Mathematics and Statistics Lanzhou University

Graph Coarsening Combinatorial Optimization

Sep 17, 2026cs.LG

COMPASS: Ordered Clustered Routing at 100K Scale

Large-scale routing often requires visiting clusters of nodes in a prescribed order, giving rise to the Ordered Clustered Traveling Salesman Problem (OCTSP). Optimizing each cluster independently seems natural, but misses non-local dependencies. We introduce the COMPASS algorithm for OCTSP, which combines search with learning-accelerated routing by orchestrating parallel sub-solvers. COMPASS has no quality ceiling and its solutions keep improving with compute. It exploits the clustered structure, and can reach exact solutions in time exponential in cluster size rather than instance size. Empirically, COMPASS consistently outperforms alternative methods. Unlike common large-scale routing solvers, COMPASS consumes general distance matrices and is not limited to coordinate inputs. We demonstrate scaling to 100K synthetic nodes and to 28.5K real e-commerce nodes. To our knowledge, the latter is the largest reported routing solution over asymmetric distances, 9x beyond established ATSP benchmarks.

Ido Greenberg, Hugo Linsenmaier, Piotr Sielski +4

NVIDIA

Traveling Salesperson Problem Combinatorial Optimization Neural Combinatorial Optimization

Sep 17, 2026cs.AI

Solving Minimum Span Antibandwidth and Cyclic Antibandwidth Labeling Problems

The Antibandwidth and Cyclic Antibandwidth problems are NP-hard graph labeling problems that aim to maximize the minimum (cyclic) distance between labels assigned to adjacent vertices. Extensive research on these problems has resulted in a variety of mathematical formulations and computational approaches. However, their minimum span perspective, in which a prescribed minimum (cyclic) distance is fixed and the objective is to minimize the label span, has received comparatively little attention. In this paper, we consider this complementary perspective by introducing the Minimum Span Antibandwidth/Cyclic Antibandwidth Labeling (MSABL/MSCABL) problems and developing a unified Boolean Satisfiability (SAT)-based framework for solving them. The SAT-based framework formulates MSABL/MSCABL as a sequence of decision problems and exploits their monotonicity to accelerate the search process. We also consider two SAT solving strategies, parallel and incremental SAT solving: the former examines multiple candidate spans concurrently, while the latter reuses a single SAT instance while progressively restricting the label domain. The proposed approaches are evaluated on benchmark instances from the Harwell-Boeing Sparse Matrix Collection and compared with CPLEXCP, CPLEXMIP, and Gurobi. The results show that SAT-based approaches are highly competitive in solution quality, with the parallel approach performing best overall for MSCABL and the incremental approach for MSABL. With the no-hole constraint, they remain competitive with CPLEXCP and significantly outperform CPLEXMIP and Gurobi, particularly for MSCABL. These results demonstrate the effectiveness of SAT solving as an exact approach for MSABL and MSCABL.

Hieu Truong Xuan, Khanh To Van

Faculty of Information Technology, VNU University of Engineering and Technology, Xuan Thuy, Cau Giay, Ha Noi, 123105, Vietnam.

Combinatorial Optimization

Sep 16, 2026cs.NE

A Metaheuristic Optimization Framework for Discrete Optimization under Strict Time Limits

Real-time applications often rely on optimization approaches that can find high-quality solutions to hard problems on the order of milliseconds. Metaheuristic optimization frameworks (MOFs) are useful tools for such tasks, as they provide large sets of general-purpose search mechanisms that can return solutions under different computational budgets. However, existing work largely overlooks the available computation time as an explicit dimension of analysis. In this work, we introduce STILO, a MOF specifically designed for optimization under strict time limits. STILO integrates fine-grained configuration spaces for ant colony optimization (ACO), genetic algorithm (GA), and simulated annealing (SA), combining existing and novel operators. We performed experiments using both synthetic and benchmark instances of various discrete optimization problems. The results indicate that the proposed discrete distance calculation mechanism for SA is useful for optimization under strict time limits. They also show that the relative effectiveness of the proposed problem-independent graph structures for ACO can vary across time limits, even for the same instance characteristics. More generally, the results demonstrate that the effectiveness of algorithm families and operators depends not only on the problem type, but also on the characteristics of the instance and the available computational budget.

Umut Çalıkyılmaz, Nitin Nayak, Sven Groppe

University of Lübeck, Lübeck, Germany · TU Bergakademie Freiberg, Freiberg, Germany

Metaheuristic Optimization Combinatorial Optimization Genetic Algorithms

Sep 16, 2026cs.RO

M $^3$ P-R1: Reinforcement Learning for Large Language Model Guided Multi-Modal Motion Planning via MIP Code Generation

Multi-Modal Motion Planning (M

^3

P) requires joint reasoning over continuous motions and discrete mode transitions, making it difficult to solve efficiently. For instance, a bipedal robot may walk to a target location and then use its arms to grasp an object. This scenario captures both mode transitions and continuous dynamics, yielding feasible paths that neither purely discrete nor continuous planners can handle. While Mixed-Integer Programming (MIP) offers a principled framework, constructing tractable formulations for non-convex problems is typically manual and domain-specific, especially in the approximate, discretization-based MIP regime needed for non-convex robotic tasks. We propose M

^3

P-R1, a reinforcement learning method that fine-tunes large language models (LLMs) to decompose M

^3

P tasks into MIP variables, constraints, and objectives. Instead of directly outputting answers, which are often prone to hallucination, the model generates executable Python code using MIP optimization libraries and constraint interfaces. This enables solver-backed execution for robust and verifiable solutions. Trained with an outcome-driven reward against the solver, M

^3

P-R1 learns to compose modality-level discretization primitives and synthesize cross-modal coupling constraints, producing executable MIP programs for complex M

^3

P tasks.

Xingpeng Sun, Zherong Pan, Kai Cheng +3

Purdue University United States · LightSpeed Studios United States

Constrained Motion Planning Large Language Model-Based Robot Planning Combinatorial Optimization Program Synthesis LLM-Guided RL Motion Planning+2