Combinatorial Optimization

The Zarankiewicz number Z(m,n,s,t) is the maximum number of edges in a bipartite graph with parts of orders m and n containing no copy of Ks,t. We give one combined, certificate-based computer-assisted proof for two finite slices and a corrected neighboring frontier: Z(12,n,3,3) = 6n (18 <= n <= 22), Z(13,22,3,3) = 137, Z(13, 18, 3, 3) = 116, Z(14, 18, 3, 3) = 124, Z(15,18,3,3) = 132, Z(14, 17, 3, 3) = 118, Z(15, 17, 3, 3) = 126, 132 <= Z(16,17,3,3) <= 133. The load-bearing new upper bounds are the exact 12 x 18 and 13 x 18 certificate packages. Their orbit certificates exclude every hypothetical matrix at the next edge count. Deletion lemmas and explicit witnesses close four neighboring cells, while the 16 x 17 entry is deliberately reported as an interval because only its 132-edge lower witness and the published 133 upper bound are certified here. Separately, the 13 x 22 proof excludes 138 ones by reducing to 83 degree profiles, rationally separating 77 of them, and eliminating the remaining six by marked-row congruences, leave enumeration, modular Gram tests, and exact Farkas certificates. All accepted claims are replayed by standard-library Python and exact integer/rational arithmetic; floating-point optimization is used only to discover certificates.

The runtime of Constraint Programming (CP) solvers is highly sensitive to modeling choices, such as symmetry breaking, implied constraints, global constraints, constraint reformulation, and variable representation. Improving these constraint models has traditionally required human expertise, and existing automated reformulation systems are restricted to a predefined library of hand-crafted transformation rules. We introduce an agentic framework that instead reformulates a constraint model from an open-ended space and establishes correctness empirically rather than by construction: a Large Language Model (LLM) agent, given a model and three training instances, proposes alternative formulations, validates each by injecting its solution back into the original model, and diagnoses and repairs failures, returning the best variant it finds in a median of about fifteen minutes. The models are expressed in the CPMpy modeling library, and each proposed model is evaluated on three larger test instances. Across nine combinatorial optimization problems, the generated models outperform the originals on 21 of 27 test instances, and on some problems solve more than two orders of magnitude faster. A comparison against non-agentic baselines that reuse the same validation and selection tools indicates that the gains stem from the agent's iterative diagnosis and repair, not merely from sampling several candidates. These results demonstrate that autonomous agentic methods can support the improvement of constraint models.

Solving combinatorial optimization problems (COPs) requires not only efficient algorithms but also carefully crafted formulations. While recent works have leveraged LLMs to automate optimization modeling, current frameworks predominantly rely on a rigid mixed-integer linear programming (MILP) paradigm. In this paper, we argue that not all problems are best modeled as MILP, as forcing complex domains into linear constraints can induce prohibitive modeling complexity and severely restrict solver flexibility. To address this, we propose OptiDSL, a framework that shifts the focus from rigid MILP formulations to domain-specific language (DSL) representations. By utilizing LLMs to map natural language onto standardized, domain-accepted structures, OptiDSL decouples problem formulation from execution. This paradigm enables seamless integration with a diverse library of specialized solvers, ranging from traditional heuristics to modern learning-based methods. Experimental results on the comprehensive benchmark of 44 COP types show that OptiDSL significantly surpasses MILP-based pipelines, yielding a 51.66% gain in formulation accuracy and a 91.71% decrease in modeling time. Notably, it also outperforms MILP-based pipelines on the existing benchmark, achieving a 23.09% higher formulation accuracy. Our code is available at https://anonymous.4open.science/r/OptiDSL.

Ising machines are emerging as promising hardware for combinatorial optimization. With recent advances in CMOS Ising technology, they are becoming attractive as low-power accelerator systems for robotics, where energy is limited and combinatorial optimization arises in multiple forms. However, a hardware-aware analysis of where such chips fit within a robotics planning stack is still missing. This paper studies the capabilities and limitations of CMOS Ising machines for low-power acceleration in multi-robot multi-target planning. We analyze three planning layers---target sharing, tour construction, and pathfinding---using real 45-spin all-to-all connected CMOS Ising chips as representative devices. We propose new Ising-based planning methods and a multi-mapping pipeline that uses spin merging, coefficient quantization, and spin-budget branching to adapt subproblems to spin- and coefficient-limited hardware. Our results show that the proposed recursive target-sharing method naturally matches the Ising hardware, achieving up to 8,000x lower energy than a classical baseline. End to end, the Ising pipeline produces routes within 9% of a strong classical baseline at 130x lower energy, showing that compact CMOS Ising machines can be effective in selected parts of the planning stack.

When a student must learn concepts connected by prerequisite dependencies, when does the order of instruction matter, and what does it cost to find the best one? We study instructional sequencing as a stochastic shortest-path problem in which attempting a concept succeeds with a state-dependent probability and failure leaves the learner state unchanged. We first prove that this stochasticity can be eliminated exactly: the problem collapses to a deterministic shortest-path problem on the lattice of prerequisite order ideals, preserving optimal values and actions. The collapse removes stochastic complexity but not combinatorial complexity: optimal sequencing remains NP-hard -- via reduction from feedback arc set in tournaments -- even with no prerequisite edges, unit costs, uniform binary nonnegative transfer, and success probabilities at least $1/2$. Hardness is not uniform: when realizable transfer preferences remain jointly acyclic with the prerequisites, any topological order of the residual joint graph is optimal, and fixed prerequisite width yields polynomial-time exact dynamic programming. A computable diagnostic, $mΔ$, bounds the value of sequencing before optimization. On 70,893 interactions from an introductory CS course, the diagnostic certifies a doubly easy regime -- little value to optimize and little space to search -- while constructed transfer instances realize the challenging regime, where myopic sequencing suffers large regret yet exact A* with a consistent heuristic expands only linearly many states on that family.

Large language model-based automated heuristic design (LLM-AHD) has shown strong potential in discovering effective heuristics for combinatorial optimization problems. However, existing methods primarily optimize a single heuristic, whereas practical optimization frameworks often rely on multiple interacting components. Directly extending single-heuristic methods is challenging because early component selection can overlook components with late potential, while independent evolution ignores inter-component dependencies. We propose MuEvo, an LLM-driven framework for evolving heuristic ensembles under ensemble-level feedback. MuEvo combines Dynamic Component Management, which uses short-budget probing and a reversible lifecycle to revise component priorities throughout the search, with LLM-Driven Co-Evolution, which coordinates component populations through Multi-Ensemble Evaluation, Cross-Component Information Sharing, Relation-Guided Pair Evolution, and Adaptive Budget Allocation. We evaluate MuEvo on selection hyper-heuristics and componentized ant colony optimization across four combinatorial optimization domains. Results show that MuEvo consistently improves human-designed frameworks and outperforms representative multi-component extensions of state-of-the-art LLM-AHD methods, demonstrating its effectiveness across both controller-mediated heuristic pools and functionally differentiated algorithmic components.

This work revealed a relationship between a multi-objective optimization problem and single-objective optimization problems that exist in the multi-objective problem. This work focused on combinatorial problems and investigated the relations between the local optima networks of the single-objective problems and the Pareto optima network of the multi-objective problem. Each of their networks has a graph structure. We divided the entire network into subgraphs. Each subgraph was called a component and characterized by overlapping relations between the single-objective local optima networks and the multi-objective Pareto optima network. Results on multi-objective landscape problems showed that most Pareto optimal solutions were reachable from the single-objective local optimal solutions. This tendency was emphasized by increasing the number of objectives and the objective correlation. The number of co-variables impacted the number of cross-link relations between the single-objective local optima networks and the multi-objective Pareto optima network. The results suggested that searching for single-objective problems is a clue to multi-objective optimization.

Deep reinforcement learning (DRL) approaches for flexible job shop scheduling (FJSP) heavily rely on attention-centric architectures to achieve state-of-the-art performance. However, these models suffer from excessive parameter counts and prohibitive inference latency as problem scales expand. While liquid neural networks (LNNs) offer a parameter-efficient alternative for modeling adaptive state evolution, their inherently sequential dynamics bottleneck computational efficiency. To resolve this trade-off, we propose PLAN (Parallel Liquid-inspired Approximation Network), a lightweight representation learning framework that reformulates continuous liquid-state dynamics into a discretized and parallelizable formulation. PLAN structurally decouples state evolution from context aggregation, where liquid-inspired updates handle the primary evolving state representation, and a lightweight context aggregation module provides complementary global context. Furthermore, PLAN acts as a versatile, plug-and-play backbone that generalizes to complex FJSP variants, pairing with a compact stochastic module for stochastic FJSP and replacing heavy heterogeneous graph transformers in multi-faceted dynamic FJSP. Extensive evaluations across deterministic, stochastic, and multi-faceted dynamic FJSP benchmarks show that PLAN reduces the average makespan by 1.2%, 1.4%, and 2.3%, respectively, compared with the corresponding state-of-the-art baselines, with the improvement reaching 10.2% in one benchmark setting. PLAN also reduces average inference latency by 13.2%, 31.7%, and 26.9%, respectively, with a maximum reduction of 69.2% on the largest instances, while using only 22$-$47% of the baseline parameters.

Delivering sustained power to distributed equipment in unstructured field environments using pre-planned wired networks or battery-based solutions presents significant infrastructure and logistics challenges. This paper presents Dendritic Recursive Pivoting (DeRP), a decentralized framework for multi-target network formation in robot swarms based solely on local communication and bearing-based sensing toward sinks. We envision a system in which robots, acting as a conduit, self-assemble a power network from a common source, forming branches at locally selected pivot points that approximate the Steiner points of Steiner trees to efficiently route to multiple Sinks. This branching operation is performed recursively to enable scalable and adaptive network formation without global planning. The proposed method is evaluated in terms of the total network length and estimated power loss, and is quantitatively compared against global baselines such as the Minimum Spanning Tree and Steiner tree solutions (GeoSteiner), which require complete knowledge of Sink locations. Specifically, we found that the networks formed by DeRP asymptotically form approximately 125% of the global minimum length while reducing power losses to 65% relative to Euclidean Steiner trees. In addition, we empirically characterize scaling behavior by measuring simulation completion time as the number of Sinks and robots increases, and find that this scaling was sub-linear for up to 100 sinks. The proposed approach enables resilient, adaptive power delivery in environments where deployment of traditional infrastructure is challenging.

Sequential decision-making in real-world applications often involves uncertainty about the environment's model. Uncertain Markov decision processes (UMDPs) represent the possible environments as a set of MDPs with shared states and actions but potentially different transition probabilities and rewards. Optimizing a single policy across all possible MDPs may sacrifice performance, while preparing an individually optimized policy for every MDP may violate operational, regulatory, or interpretability constraints on the number of policies that can be prepared and deployed. We consider settings in which model uncertainty is resolved shortly before execution, allowing the most suitable policy to be selected from a limited set prepared in advance. We introduce $k$-adaptable policy synthesis, which optimizes such a set of $k$ policies under a minimax-regret objective. We prove that the problem is NP-hard and develop KAPS, an exact nested branch-and-bound algorithm with problem-specific bounds and heuristics. KAPS jointly optimizes which MDPs share a policy and the policies themselves. Experiments across various UMDP benchmarks show that the largest reduction in regret consistently occurs when increasing from one to two policies. In the single-policy setting, KAPS is competitive with existing methods in solution quality and proves optimality substantially more often.

We study the problem of optimizing physical layouts for automated warehouses, where hundreds to thousands of robots are coordinated to transport packages. Previous works have shown that optimizing the warehouse layout (e.g., the physical location of the storage shelves) significantly improves throughput. However, state-of-the-art layout optimization approaches are based on evolutionary optimization methods, which treat the entire warehouse as a black box and rely on random mutation to search for high-quality layouts. While the optimization outcomes are promising, these methods require a massive number of simulations to evaluate candidate solutions, making them sample-inefficient. In this paper, we present Stress-Relief Annealing (SRA), a polynomial-time simulation-free layout optimization algorithm. SRA turns the task demand into a per-vertex \emph{stress field} that predicts where traffic will concentrate in the warehouse; the field's peak provably caps the throughput. Our experimental results show that (1) SRA improves both the throughput and the scalability of a human-designed warehouse, roughly doubling the number of robots it can sustain, (2) it matches or exceeds the throughput of the evolutionary baselines while taking only $19$ minutes on one CPU core, against their $25{,}000$ simulations and $25$ hours on a $64$-core machine, and (3) the gain generalizes across different Multi-Agent Path Finding algorithms, non-uniform task demands, and a warehouse with doubled dimensions.

This paper introduces SCHEDBench, a natural-language benchmark for evaluating combinatorial scheduling constraint faithfulness under surface-form variation. Grounded in canonical scheduling instances and solver-derived feasibility and optimality, SCHEDBench assesses whether large language models (LLMs) generate schedules with the same constraint-feasible behavior across varied natural-language (NL) surface forms. SCHEDBench spans 1,132 instances across job-shop scheduling problems (JSP), single and multi-mode resource-constrained project scheduling problems (RCPSP), nurse rostering/scheduling, and curriculum timetabling problems of varying difficulty. Instances are templated into natural language problems using domain-specific templates, themed entities, lexical-syntactic template rephrasing, and constraint-level surface-form variation, with reference solutions verified for feasibility and objective optimality. Across thirteen frontier and open-weight LLMs, we find that models are not reliably invariant to semantically equivalent renderings of the same scheduling problem. Surface-form variation reduces feasibility and induces above-noise shifts in per-instance hard-constraint violations on matched instances. Among the tested isolated axes, constraint reordering yields the clearest above-noise sensitivity.

The rapid development of Large Language Models (LLMs) has opened new avenues for Automated Heuristic Design (AHD) for solving NP-hard combinatorial optimization problems (COPs). However, existing LLM-driven AHD methods are largely confined to rigid solver templates, relegating the search process to isolated module tuning. Transitioning to fully autonomous, system-level algorithm design is essential but fraught with low reliability of generated operators, extremely large search spaces, and ineffective credit assignment. To overcome these drawbacks, this paper proposes a Directed Graph-Guided Automated Algorithm Design framework, termed DGA$_2$D. It structures the open-ended program space as a directed graph, where each node represents a functional operator that can be instantiated using one of multiple candidate code implementations, while directed walks constitute complete algorithmic pipelines. A first-order path-dependent credit assignment mechanism is introduced to evaluate code variations strictly based on their topological context. Extensive experiments across 12 distinct COPs, ranging from complex scheduling to routing, demonstrate the consistent empirical advantages of DGA$_2$D. It reduces the average normalized gap by up to 10.96 percentage points compared to state-of-the-art LLM baselines.

Leader Reward modifies POMO training to emphasize the best trajectory produced by repeated inference. We test a narrow extension: replace its binary leader/non-leader distinction with a stabilized rank signal indexed by a sampling budget $K$. With the POMO architecture, 3,050-epoch schedule, and TSP-100 test set held fixed, the Leader Reward reimplementation obtains $7.7662$ under 100-start, 8-augmentation greedy decoding, matching the reported $7.766$ at its displayed precision. Under independent sampling, the stabilized $K=8$ recipe lowers realized Best-of-8 cost in all three paired training seeds: $7.7944$ versus $7.8136$. This observation is estimation-only and decoder-specific: three seeds are below the six-seed testing floor, Leader Reward is better at sampled $K=1$, and it remains slightly better under its original augmented-greedy protocol. We make no unbiased-estimator, universal superiority, or state-of-the-art claim.

Automated heuristic design (AHD) with large language models (LLMs) has produced strong heuristics for combinatorial optimization problems (COPs). Yet existing frameworks optimize for average performance on a small fixed dataset and steer the search with "verbal gradients" distilled from scalar better/worse feedback. No single heuristic dominates across instance distributions, and scalar feedback tells the LLM whether a heuristic improved, but not where in the instance space or why. We propose MOSAIC, a grid-based framework that adversarially co-evolves problem instances and specialist heuristics inside a Quality-Diversity (QD) archive indexed by structural instance features. Instances evolve to expose weaknesses of the current heuristics, and heuristics evolve to eliminate them by specializing to the newly exposed regions. Each archive cell keeps a specialist heuristic, representative instances, and insights explaining what works in its region, forming a persistent memory that accumulates over the evolutionary search. For each heuristic pair sampled from distant grid regions, an LLM-guided evolutionary loop generates discriminative instances, and a decision tree identifies the feature-space regions where each heuristic wins. A reflection LLM then contrasts the two heuristics to produce multi-directional insights that persist in those regions and guide crossover and mutation. The archive is simultaneously a co-evolved benchmark of discriminative instances and a pool of region specialist heuristics, from which greedy selection extracts a compact complementary portfolio. Across COPs, test sizes, and LLM backbones, the portfolio consistently outperforms state-of-the-art LLM-based AHD methods, and the co-evolved instances attain higher feature-space coverage and stronger heuristic discrimination than evolutionary instance-generation baselines.

Machine learning for combinatorial optimization typically relies on neural constructors trained via reinforcement learning on large offline datasets for a fixed problem class-incurring high pretraining costs and generalizing poorly outside the training distribution. We propose an alternative: a metaheuristic framework that reformulates the randomized constructive phase of GRASP as an online imitation learning task, trained from scratch on each problem instance. A local search procedure acts as an expert oracle, while a decoder-only Transformer serves as the constructive policy. Unlike classical GRASP, which relies on static, myopic heuristic rules based on localized scalar costs, our approach is fully data-driven: the construction policy emerges from high-quality solutions discovered during the search itself, with no problem-specific feature engineering required. We instantiate this as LM-GRASP, a hybrid metaheuristic following an iterative learn-infer-improve cycle, training the policy online via behavioral cloning on a dynamic archive of elite trajectories-no external data or offline pretraining needed. The pipeline interfaces with the domain solely through the objective evaluator used by local search. Evaluated on the Taillard PFSP benchmark (ta51-ta60), the most discriminating block due to half its optima being unknown, LM-GRASP outperforms GPU-GRASP by 28.4 makespan units on average-comparable to the gain from GPU acceleration over sequential execution (27.2 units), though with overlapping standard deviations. This suggests instance-specific, online-trained language models are a promising, practical alternative to hand-engineered constructors, especially for landscapes resistant to classical greedy construction.

In dynamic multi-mode project scheduling, activities have alternative execution modes and uncertain durations, while precedence relations and limited resources constrain their execution. Heuristic priority rules support fast online decisions, but their design requires substantial domain expertise. Genetic programming (GP) hyper-heuristics can automatically evolve such rules. Large language models (LLMs), meanwhile, provide a flexible interface for interpreting scheduling information and explaining decisions. However, zero-shot LLM decisions may lack domain knowledge, consume many tokens, and vary across repeated queries. GP-evolved rules therefore provide a potential source of scheduling knowledge for guiding LLM decisions. Unlike existing LLM--GP hybrids that use LLMs to support heuristic evolution, we transfer knowledge in the reverse direction, using knowledge extracted from high-quality GP rules to guide an online LLM decision maker. We extract knowledge from high-quality GP rules and inject it through Feature Selection, Feature Hint, Rule Reference, and Rule Follow. These mechanisms are evaluated in terms of scheduling performance, token consumption, decision stability, and the feature focus expressed in generated rationales. GP-derived guidance generally improves the unguided LLM, but its representation matters. Simplifying the decision context or supplying explicit decision logic is more effective than highlighting important features. Feature Selection offers the best token efficiency, whereas Rule Follow achieves strong performance at greater token cost. Guidance also improves decision stability and changes the features expressed in generated rationales.

This paper introduces the edge-based contiguous p-median (ECpM) problem to partition the roads in a network into a given number of compact and contiguous territories. Two binary programming models are introduced, both of which incorporate a network distance. The first model requires an exponential number of cut set-based constraints to model contiguity; it is paired with a separation scheme that usually generates only a small number of these constraints, namely, a branch-and-cut (B&C) algorithm. The second model utilizes a polynomial number of shortest-path constraints to model contiguity and can be solved with off-the-shelf solvers. The respective solution approaches are tested on road networks with over 2,700 nodes and close to 3,400 edges, yielding models with over 9.6 million binary variables. Solving the model based on shortest path contiguity (SPC) constraints via standard branch and bound attains speedups in computational time of up to 17x relative to the cut set-based B&C implementation. In addition, the SPC constraints are demonstrated to be supervalid inequalities of the edge-based p-median (EpM) model (i.e., for which contiguity is not explicitly required), meaning that they may cut off integer-feasible solutions and some, but not all, of the optimal solutions of this simpler problem. Finally, the paper explores structural insights and connections between ECpM and the edge-based districting (EBD) problem, which enforces an additional work balance criterion. An existing model that utilizes cut set-based contiguity constraints was unable to find a feasible solution within 12 hours for any of the tested instances, while an SPC-based EBD model was able to solve most of these to optimality.

Black-box combinatorial optimization requires systematically identifying high-quality solutions under a limited evaluation budget, yet the unknown objective function provides little guidance for deciding where the search should explore next. We introduce SCOPE, a general framework for Synthetic Conditional Objectives for Policy Evolution in Black-Box Combinatorial Optimization. Rather than directly optimizing the inaccessible objective, SCOPE learns a set of synthetic objectives conditioned on the accumulated search history, where each objective is designed to expose a distinct and potentially useful preference over candidate solutions. These objectives are then used to evolve search policies that generate diverse candidates, whose true quality is subsequently assessed through black-box evaluations. The outer loop adaptively updates and selects synthetic objectives according to how effectively their induced policies discover promising regions. In contrast, the inner loop returns a portfolio of top-performing policies to reduce the risk of relying on a single surrogate preference. This formulation reframes objective design as a mechanism for guiding policy exploration, enabling the search process to exploit observed evidence while maintaining structured diversity across discrete solution spaces. Extensive experiments across multiple benchmark problems demonstrate that SCOPE consistently improves black-box search performance under limited evaluation budgets and generalizes well across diverse combinatorial structures.

Multi-agent planning problems arise in a variety of engineering applications, such as multi-robot wildfire fighting and unmanned aerial inspection in factories. A particular challenge is the existence of spatio-temporal (i.e., when and/or where an agent should do what) and topological constraints (i.e., how agents should interact), as typically formalized via the notion of graphs. Over the last years, various frameworks have been proposed that can capture such constraints via spatio-temporal logics. We focus here on spatio-temporal logic with graph operators (STL-GO), a recent formalism that supports reasoning about multiple agents and their topologies, such as sensing, communication, and task topologies. In this paper, we consider the problem of planning multi-agent paths that satisfy constraints written in STL-GO. This problem is particularly challenging due to the need of encoding multiple, potentially time-varying graphs via the graph operators inherent to STL-GO. We present two encodings of this problem, one based on mixed-integer programming (MIP) and another based on satisfiability modulo theory (SMT), with soundness guarantees. We provide a unified interface for specifying agent constraints, their graph topologies, and the STL-GO specification, enabling seamless use of both methods and facilitating direct comparison between them. We evaluate both encodings on a multi-UAV search-and-rescue benchmark, ablating over team size and graph complexity, highlighting the expressiveness of the proposed encodings under dynamic multi- graph interactions.

In some real applications a plan may later become unfeasible due to newly imposed budget constraints, yet, at the same time, using only the original actions of the plan and their order is mandatory. In this paper, we study the problem of extracting, from a precomputed plan, a valid subplan that maximizes utility while respecting a cost bound. Each goal is given a utility value and the plan is reduced by removing actions that support low-utility goals, while preserving both executability and the original action order. We show the decision variant is NP-complete and propose two exact methods to solve it: one via oversubscription planning (OSP) and another via Integer Linear Programming (ILP). This paper extends our previous work published at ICAPS 2026 (Del Toro, Fuentetaja, and García-Olaya 2026b). While the core framework remains as introduced there, we further introduce a refined ILP formulation that significantly decreases the model size and improves computational efficiency.

This study presents a theoretical analysis of partitional clustering on networks, analyzing both hard and soft assignment schemes with different objective functions. Cluster centers are not restricted to vertices but can also be located along the edges. We examine four key models: P-Median (PMP) and Sum of Squares Clustering (SSC) under hard assignment, and Probabilistic Distance Clustering (PDC) and Fuzzy C-Means (FCM) under soft assignment. Through mathematical analysis, we uncover structural properties that differentiate these models, such as the significance of assignment bottleneck points and the role of vertex-restricted solutions in determining optimal cluster centers. Our findings reveal that, while SSC and FCM can yield optimal centers along edges, PMP and PDC inherently favor vertex placement, leading to insights into clustering behavior on networks. These insights offer new directions for designing efficient algorithms and have implications ranging from facility location and network design to clustering on the embedding graphs that power similarity search in modern retrieval systems.

Humans often find good solutions to combinatorial optimization problems that are computationally hard even for advanced computer algorithms. In the Euclidean traveling salesman problems (TSP), people rapidly produce tours that are near-optimal, despite severe limits on time and computation. What makes a tour human-like, and how might such solutions be learned? Here we address these questions through a large-scale behavioral and computational investigation of human performance in Euclidean TSP. We sampled a broad space of TSP instances, collected human solutions, and compared them with neural policies based on Pointer Networks, which are recurrent neural networks with an attention-based pointing mechanism that define probability distributions over valid tours. We trained these networks under multiple objectives, including reinforcement learning (RL), supervised learning from optimal tours, supervised learning from human tours, and RL fine-tuning after optimal-supervised pretraining. Human tours were not identical to optimal tours, but occupied a near-optimal geometric basin: they shared many structural properties with optimal solutions while preserving systematic human-specific deviations. The best account of human tours was not direct imitation of optimal tours, but a model pretrained on optimal tours, fine-tuned by RL, and decoded through $\text{Best-of-}N$ sampling. These findings suggest that human-like solutions may emerge from a combination of structured supervised learning, RL, and test-time search, echoing computational principles underlying many modern artificial intelligence systems.

LLM-based automated heuristic design (AHD) typically scores executable programs on complete instances or within fixed solver components. In large-scale routing problems, localized reconstruction reduces the size of each optimization task, but repair regions within the same incumbent can exhibit substantially different structures. One construction rule must therefore compromise across them. In this paper, we propose SpecAHD, a coupled bilevel framework for within-instance specialization. An upper-level search learns where to expose bounded repair regions, while a lower-level search evolves a complementary repertoire of executable heuristics for the induced repair tasks. The upper-level program determines the repair tasks seen by the lower level, while checked repair outcomes determine how upper-level programs are evaluated. The lower-level objective favors heuristics that perform well on average or solve tasks that the current repertoire handles poorly. For the repair tasks induced by a fixed upper-level program and a fixed lower-level candidate pool, this objective is monotone submodular, allowing greedy repertoire selection with a (1-1/e) approximation guarantee. Across four routing problems and multiple LLM backbones, SpecAHD reduces held-out objective cost by up to 57.7% against the strongest competing AHD baseline and outperforms the per-instance baseline envelope on most public instances.

We study an integrated pickup-and-delivery problem on sparse, non-Euclidean networks that jointly optimizes cyclic routing, cargo flow allocation, and cross-cycle service. The tight coupling of these operational constraints creates a complex discrete-continuous decision space with highly restricted feasible regions. To overcome these computational challenges, we propose Double-Channel Graph Attention (DCGA), an end-to-end reinforcement learning framework. DCGA isolates network reachability and demand-service logic into separate graph channels and constructs valid routes using a simulator-coupled, constraint-informed decoder. Experiments on LinerLib benchmarks demonstrate that DCGA achieves seconds-level inference and delivers state-of-the-art solution quality on instances beyond a specific scale, with its advantage over existing baselines widening significantly as problem size increases. Supported by extensive stability and ablation analyses, our results demonstrate that this structure-aware learning approach provides an effective, low-latency engine for realistic routing-and-flow optimization.

The optimal allocation of academic resources to individual students is essential for addressing learner diversity and fostering equitable educational outcomes. Within the framework of the Erasmus+ KA220-SCH project, this paper models the selection of educational materials for high school mathematics students as a 0--1 binary combinatorial optimization problem subject to strict study time constraints. Given the NP-hard complexity of the formulation, exact solution methods become computationally intractable as resource catalogs scale. To address this challenge, we propose a Genetic Algorithm integrated with a specialized constraint repair mechanism to effectively search the binary decision space. Experimental evaluation across 10 independent runs demonstrates fast convergence, high solution quality, and strong algorithmic stability across different base seeds. These results confirm the practical utility of metaheuristic approaches for real-time decision-support systems in secondary education.

Reusing previously computed results is a long-standing principle for reducing computational cost, but such reuse has largely been confined to a single problem's computation. Sharing computational processes across multiple simultaneously solved problems remains possible in principle, yet designing algorithms that exploit nontrivial cross-task relationships is difficult to do manually. Here, we use machine learning to discover such algorithms automatically. Specifically, based on reservoir computing, we propose a method that uses computation results recorded by dynamic programming for combinatorial optimization problems as features for linear regression, leveraging them to assist other combinatorial optimization computations. We validate the approach on the traveling salesman and subset sum problems. Multiplexing the dynamic programming process improves approximation accuracy over generic features and reduces computation time compared with independent solutions. These results suggest a new form of computation, distinct from conventional computational design, in which multiple processes efficiently share and recycle intermediate results and states.

The Shannon capacity $Θ(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $α(G^d)^{1/d}$ for any $d$, where $α(G^d)$ is the independence number of the $d$-th strong product of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, $62530$ in $C_{13}^{6}$, and $8076974$ in $C_{15}^{8}$, improving the best known lower bounds for the Shannon capacity of these graphs to $Θ(C_7)\geq 134753^{1/10}>3.258020$, $Θ(C_{11})\geq 21909^{1/6}>5.289773$, $Θ(C_{13})\geq 62530^{1/6}>6.300109$, and $Θ(C_{15})\geq 8076974^{1/8}>7.301399$. We also improve the best known lower bounds on the independence numbers of several individual strong products of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.

Multi-object tracking in dense crowds requires solving a bipartite assignment problem between detections and trajectories at every video frame. The classical Hungarian algorithm solves this in $O(N^3)$ time, which becomes a bottleneck for large scenes with hundreds of people. We propose an \emph{incremental} assignment algorithm that exploits the block-sparse structure of crowd tracking cost matrices --- dense within each crowd cluster, near-zero between clusters. We compute the exact same optimal $N \times N$ assignment as the Hungarian algorithm, but via an incremental strategy: we add one person at a time, exploiting the fact that after step $n-1$ the dual potentials are \emph{exactly optimal} for the $(n-1)\times(n-1)$ subproblem --- a strictly stronger condition than the intermediate feasibility maintained by the Hungarian algorithm during its $N$ outer iterations. Each new step therefore requires only a single augmenting path search from a certified optimal starting point. This avoids repeated full-matrix scans while guaranteeing an identical globally optimal result. A diagonal-reordering invariant keeps the data structure compact and cache-friendly. On realistic crowd benchmarks with $N \in [200, 5000]$ people organised into dense clusters, our algorithm achieves \textbf{3.7--6.5$\times$ speedup} over the Hungarian baseline while producing provably optimal matchings identical to those of Hungarian. The speedup grows with $N$ and remains stable beyond $N=3000$, making the method especially attractive for large-scale crowd scenes such as stadium exits and mass public events.

For years, supply chain planning at e-commerce firms has operated as a collection of isolated projects. Each planning task from static network planning to dynamic warehouse assortment planning requires analysts to spend weeks building models from scratch, calibrating and persuading executives to act on outputs they cannot verify. Three barriers drive this: bespoke models proliferate because standardization is difficult (operational fragmentation); once unified, the combinatorial scale of millions of SKUs, thousands of nodes, and intricate routing logic exceeds what solvers can handle within a tight window (computational intractability); and a mathematically optimal solution still fails to be implemented if the executives do not trust it (implementation hurdle). To bridge this gap, we propose and implement the Simulation-Propose-then-OR-Dispose method, deployed as JD.com's NetSim platform. The central insight is decoupling: simulation proposes by generating and evaluating the full set of operationally valid candidate paths-absorbing all idiosyncratic business logic, while an integer program disposes by selecting the globally optimal subset. Computationally, matrix-vectorized CPU/GPU accelerated simulation achieves a 10-100 times speedup over serial methods, and a list scheduling algorithm reduces coupled-order processing from hours to minutes. Operationally, we establish a closed loop via an intelligent diagnosis engine. Since 2025, NetSim has optimized end to-end services for over 20,000 suppliers, the cross-regional fulfillment rate dropped from 6.1% to 4.9%, and the average monthly carbon reduction is approximately 5,745 tCO2e. SPORD moves simulation from monitoring to active planning. The transparent outputs turn skeptical executives into engaged collaborators, and the modular architecture ensures that the next planning requires just configuration, not reconstruction.