formulate combinatorial optimization problems

This work addresses the limitation in combinatorial optimization where local search neighborhoods typically require manual construction. It proposes, for the first time, a method that automatically generates functional neighborhoods by exploiting symmetries present in constraint specifications. By integrating constraint programming, symmetry analysis, and local search techniques, the approach enables automated neighborhood construction within the IDP system, substantially reducing the need for human intervention. Empirical evaluation across six classical optimization problems demonstrates the effectiveness of the generated neighborhoods, confirming both the feasibility of the method and its capacity to enhance the automation and generality of local search algorithms.

This paper investigates whether the branch-and-bound (B&B) algorithm exhibits polynomial-time approximation scheme (PTAS) behavior for NP-hard combinatorial optimization problems, including the knapsack and scheduling problems. Through structural problem analysis, novel truncation strategy design, and rigorous convergence analysis, we establish—for the first time—theoretical guarantees that the standard B&B framework asymptotically generates (1−ε)-approximate solutions within polynomial time. This result fundamentally challenges the conventional view that B&B only ensures eventual optimality, and instead bridges B&B with approximation algorithms by formally extending its theoretical applicability to polynomial-time approximation. Extensive experiments on benchmark instances confirm that the proposed approach achieves arbitrary approximation accuracy ε > 0 in polynomial time, matching or surpassing the performance of specialized PTASs and state-of-the-art heuristic methods.

This paper addresses systemic bias against social subgroups—particularly minority groups—induced by vertex cover and feedback vertex set algorithms on real-world graphs in combinatorial optimization. We propose the first modeling paradigm that explicitly incorporates group fairness into the objective function. Unlike conventional approaches optimizing only for total cost, our framework introduces a weighted graph model annotated with group labels, designs approximation algorithms satisfying explicit group-fairness constraints, and establishes a quantifiable bias measurement framework. Theoretical analysis guarantees bounded approximation ratios under fairness constraints. Extensive experiments on diverse real-world and synthetic graphs demonstrate that our method reduces inter-group disparity in solution impact by 40–65%, while incurring only a marginal increase in total cost (<15%). This yields substantially improved algorithmic fairness and societal applicability without compromising computational efficiency.

This paper addresses three NP-hard combinatorial optimization problems: the α-neighborhood p-median problem, tree-shaped hub location, and node-capacitated graph partitioning. To tackle them uniformly, we propose a generic Random-Key Optimization (RKO) framework. RKO employs a modular random-key encoding scheme coupled with problem-specific decoding mechanisms, integrates multiple parallel metaheuristics—including simulated annealing, iterative local search, and GRASP—and coordinates search via an elite solution pool. Its novel plug-and-play architecture enables rapid cross-domain adaptation. Implemented efficiently in C++, RKO consistently delivers high-quality solutions across all three problem classes, significantly enhancing both robustness and generalization capability. The framework establishes a scalable, unified paradigm for solving diverse NP-hard combinatorial optimization problems.

To address two key bottlenecks in leveraging large language models (LLMs) for combinatorial optimization problems (COPs)—weak generalization of search direction and high evaluation overhead—this paper proposes a dual-mechanism framework: Core Abstraction Prompting (CAP) and Prototypical Performance Prediction Prompting (PPP). CAP introduces the first abstraction of elite heuristics’ core components into transferable prompt priors; PPP enables zero- or few-shot heuristic performance prediction via semantic similarity, augmented by reliability discrimination and accuracy enhancement modules. Integrated with theoretically grounded prompt optimization, heuristic semantic equivalence identification, and a multi-LLM collaborative evaluation framework, our approach achieves state-of-the-art performance across four COP task categories, five canonical problems, and eight LLMs. Hercules-P, our implementation, significantly reduces evaluation cost. Ablation studies confirm the effectiveness of each component.

Variable selection heuristics in branch-and-bound (B&B) for mixed-integer linear programming (MILP) suffer from low efficiency and poor generalization. Method: We propose the first model-based reinforcement learning (MBRL) framework for B&B, which learns a dynamic environment model of the B&B search process and integrates Monte Carlo tree search (MCTS) for forward-looking, adaptive branching decisions—yielding both interpretability and sample efficiency. Contribution/Results: Our approach overcomes the dual limitations of static heuristics and model-free RL in modeling capacity and data efficiency. Evaluated on four standard MILP benchmarks, it consistently outperforms state-of-the-art RL-driven branching policies, achieving significant reductions in solving time, number of explored nodes, and optimality gap. These results validate the effectiveness and scalability of model-guided planning for combinatorial optimization.

This work addresses the limitations of existing automated heuristic design approaches, which predominantly rely on bottom-up code search and struggle to extract reusable, transferable high-level knowledge. The authors propose a top-down, knowledge-first search paradigm that treats knowledge as the primary object of search, using code merely for instantiation and validation. By formulating explainable hypotheses, the method enables knowledge reuse across problems and solution trajectories. It establishes, for the first time, a bidirectional bridge between knowledge and code, introduces a statistical learning perspective to characterize the distortion–compression trade-off, and integrates large language models with population-based and tree search mechanisms into a unified, knowledge-driven iterative optimization framework. Experiments demonstrate that this approach significantly outperforms code-centric methods in heuristic discovery efficiency, transferability, and generalization across combinatorial optimization and extended tasks.

Traditional combinatorial space representations—such as integer or binary encodings—introduce spurious relationships, dimensional inflation, and extraneous constraints in mixed-combinatorial nonlinear optimization, thereby degrading search efficiency. This work proposes a direction-aware directed graph abstraction that leverages an Edge Field Graph Network (EFGN) to map an undirected fully connected combinatorial graph into a structured directed improvement-direction graph. This graph is embedded within the optimization framework as a recommendation system, enabling search exclusively over continuous variables while dynamically retrieving optimal combinatorial configurations. The approach achieves, for the first time, a scalable and interpretable structured modeling of combinatorial spaces. Evaluated on three nonlinear benchmark problems, it significantly outperforms index-based combinatorial baselines, yielding superior average solutions and enhanced robustness.

This work addresses the computational intractability of large-scale combinatorial optimization problems arising from their exponentially sized search spaces by proposing a structure-aware parallel decomposition framework. The approach constructs a constrained maximum-cut model based on variable interaction structures, reformulates it as a QUBO problem, and leverages an Ising machine to efficiently cluster variables for automatic problem decomposition. The resulting subproblems are then solved in parallel using mathematical optimization solvers. This method uniquely integrates structure-aware clustering with Ising-based computation, substantially reducing the effective problem size. Experimental results on the capacitated vehicle routing problem demonstrate up to a 95.32% reduction in variable count, achieving within one minute the solution quality that conventional methods require thirty minutes to attain, while significantly improving the rate of feasible solutions.

This work investigates the implicit structural representation capability of large language models (LLMs) for combinatorial optimization problems and its utility for downstream decision-making tasks—specifically solver selection. Methodologically, we systematically analyze how hidden layers of LLMs encode problem structure across four benchmark optimization problems and three instance encodings, employing both direct prompting and neuron probing techniques. Our key contributions are threefold: First, we provide the first empirical evidence that intermediate LLM layers capture optimization problem structures highly aligned with classical hand-crafted features. Second, these implicit representations achieve solver recommendation accuracy on par with traditional feature-engineering approaches at the instance level. Third, the representations exhibit strong cross-problem generalization robustness. Collectively, these findings offer new insights into the internal mechanisms of LLMs in symbolic reasoning tasks and advance their trustworthy deployment in operations research and optimization.