solve constrained combinatorial optimization

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.

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 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.

In real-world scenarios, enumerating all optimal or feasible solutions to combinatorial optimization or constraint satisfaction problems is often required for informed decision-making; however, deterministic algorithms suffer from combinatorial explosion and poor scalability. This paper proposes a sampling-based solution enumeration framework tailored for physical Ising machines: the problem is formulated as an Ising model, and controlled probabilistic sampling is employed to explore the energy landscape. Crucially, we introduce the first theoretically grounded stopping criterion with statistical guarantees—ensuring the probability of incomplete enumeration remains below a user-specified threshold. By embracing the intrinsic stochasticity of Ising hardware, our approach departs from deterministic paradigms. Evaluated on maximum clique enumeration, it significantly outperforms specialized branch-and-bound algorithms, enabling efficient and complete enumeration of all maximum cliques in large-scale dense graphs.

The absence of a universal, domain-agnostic dynamic programming (DP) modeling paradigm hinders systematic DP application to combinatorial optimization. Method: This paper introduces Domain-Independent Dynamic Programming (DIDP), a paradigm that decouples problem modeling from solving. We design DyPDL—a formal language for specifying DP models—and develop CAASDy, a general-purpose solver that unifies classical DP, A* search, and cost-algebraic state-space search within a verifiable DP framework for the first time. CAASDy supports interoperable interfaces with MIP and CP models, enabling fair empirical comparisons. Results: Experiments across multiple standard combinatorial optimization benchmarks demonstrate that CAASDy significantly outperforms leading commercial MIP and CP solvers. These results validate DIDP’s triple innovation: modeling generality, solving efficacy, and theoretical verifiability.

This work addresses the modeling limitations in pathfinding tasks arising from the tight coupling between problem graphs and movement graphs. To overcome this, the paper introduces a directed, weighted bipartite graph model that explicitly decouples feasibility—defined by the problem graph—from movement rules—governed by the movement graph—for the first time. This formulation naturally accommodates asymmetry, heterogeneous constraints, and weighted transitions, thereby unifying and extending classical solution-discovery frameworks. Leveraging tools from combinatorial optimization, graph theory, and computational complexity, the study provides a complete characterization of the complexity landscape for both general pathfinding and shortest-path problems under the proposed model, precisely delineating polynomial-time solvable cases from those that are strongly NP-hard.

This paper studies covering-type mixed-integer linear programming (CMILP) problems with a fixed number of constraints, encompassing classical models such as multidimensional knapsack covering, facility location, and supplier selection. Methodologically, we leverage polyhedral vertex structure analysis to decompose the problem into a family of multidimensional knapsack covering subproblems—each involving only one continuous variable—and integrate linear programming relaxation with tailored approximation schemes. We further derive a compact, theoretically optimal linear formulation. Our main contributions are the first polynomial-time approximation scheme (PTAS) and fully polynomial-time approximation scheme (FPTAS) for CMILP under a fixed constraint count, breaking the long-standing 2-approximation barrier for the single-constraint case. Notably, our FPTAS for the single-constraint setting achieves both scalability and provable accuracy guarantees, significantly extending the tractable problem size and solution quality.

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 addresses extremal point configuration problems in combinatorial geometry under strict global constraints such as the No-Three-in-Line (N3IL) condition, where traditional approaches suffer from combinatorial explosion and sparse rewards. The authors propose the first geometry-aware Monte Carlo Tree Search (MCTS) framework that directly embeds geometric constraints into the action space, enabling incremental O(n²) collinearity verification. By integrating symmetry-based pruning and batched state transitions, the method achieves substantial computational efficiency gains. It establishes new best-known results on five of six classical benchmarks: constructing Max-N3IL configurations of size up to 1.8n on grids with 82 ≤ n ≤ 119, and reducing the upper bound for the Smallest Complete Set problem to approximately 0.95n, thereby significantly enhancing scalability and solution capability.