column generation

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 paper studies the minimization of separable convex functions over integer feasible sets defined by constraint matrices with small coefficients and bounded primal/dual tree depths. For this class of sparse integer programs, we present the first near-linear-time algorithm for nonlinear separable convex objectives that matches information-theoretic lower bounds. Our method introduces a unified framework integrating scaling techniques, proximity analysis, sensitivity theory, and dynamic data structures. When parameterized by the primal tree depth, the algorithm achieves the optimal time complexity $O(n log |u-l|_infty)$. When parameterized by the dual tree depth, it runs in $O(g n log n log |u-l|_infty)$ time—nearly matching the conjectured optimal bound. These results substantially extend the frontier of efficient solvability for convex integer programming, particularly for structured sparse instances.

This paper investigates the computational complexity of sum-of-squares (SOS) relaxations in copositive programming. Focusing on standard quadratic programming and its reciprocal problem, it establishes the **first exact sufficient condition** under which SOS relaxations are solvable in polynomial time, and characterizes the boundary of infeasibility; when this condition fails, pathological instances arise where solution size exhibits double-exponential blowup. Methodologically, the work integrates the SOS hierarchy, semidefinite programming (SDP), and the ellipsoid method, and introduces a weighted modeling framework for the stability number. Key results show that the SOS bound is polynomial-time approximable, and the (weighted) stability number of a graph admits an efficiently computable SOS upper bound. This work provides the first systematic characterization of the tractability boundary and intrinsic computational nature of SOS methods in copositive optimization.

This paper investigates the parameterized tractability of block-structured integer programming—specifically two-stage stochastic IPs and n-fold IPs—when the global constraint matrix contains arbitrarily large entries. Addressing the limitation of prior work requiring all input coefficients to be bounded, we establish, for the first time, that feasibility checking for two-stage stochastic IPs and linear optimization for *uniform* n-fold IPs are fixed-parameter tractable (FPT) when parameterized solely by the local matrix dimensions and the maximum absolute value of the right-hand sides (D_i). We further show that uniformity is necessary for n-fold IPs to admit such FPT algorithms. Our approach integrates Graver basis theory, potential function analysis, dynamic programming, integer conic decomposition, and distance-sensitive search to design a weakly polynomial-time FPT algorithm. This work establishes the first FPT framework for block-structured IPs tolerant of large matrix entries, tightening the parameter dependence from global coefficient magnitude to local magnitudes—thereby substantially broadening the applicability of parameterized algorithms for structured integer programming.

For the NP-hard Job Shop Scheduling Problem (JSP), this paper proposes a decomposition-and-coordination framework based on dynamic time-window partitioning and multi-round Answer Set Programming (ASP) solving. The method decomposes the global scheduling task into temporally coupled subproblems via multi-granularity time-window segmentation, then iteratively refines solutions using ASP. Its key contributions are: (i) the first introduction of overlapping and adaptive time-window compression mechanisms; and (ii) integration of operation-selection heuristics with window-based decomposition to mitigate local optima arising from sequential optimization. Experimental evaluation on standard JSP benchmarks demonstrates that, compared to single-shot full-scale ASP solving, the approach significantly improves both solution efficiency and optimality under tight time limits. Moreover, applying window-based refinement to heuristic-initial solutions reduces average makespan by 12.7%.

This work addresses the computational challenges of solving large-scale convex mixed-integer quadratic programs (MIQPs), which arise in applications such as subset portfolio selection and become particularly difficult when the covariance matrix has a high condition number or weight constraints are tight. To tackle this, the authors propose DASH, a novel method that introduces a decreasing active-set hierarchy for dimensionality reduction in MIQP for the first time. DASH leverages active-set analysis to reduce problem dimensionality and integrates seamlessly with commercial solvers like Gurobi to enhance optimization efficiency. Experimental results demonstrate that DASH significantly outperforms Gurobi alone on a range of challenging portfolio instances, with solution quality improvements positively correlated with problem difficulty, thereby accelerating convergence and yielding higher-quality optimal solutions.

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 proposes an efficient method for atomic decomposition of undirected graphs that circumvents the complex pipeline traditionally reliant on minimal triangulation and minimal clique separators. For the first time, it establishes a theoretical connection between Maximum Cardinality Search (MCS) orderings and atomic decomposition, enabling the direct recursive generation of atoms through the convex extension of node 1 and its neighborhood. By integrating two existing convex extension strategies, the proposed algorithm achieves significantly improved computational efficiency while preserving correctness. Experimental results demonstrate that the method outperforms current state-of-the-art graph decomposition algorithms in terms of runtime performance.

This study addresses the feasibility of leader-induced follower responses in bilevel matching optimization, where the leader specifies mandatory and forbidden edges. Focusing on maximum-weight matchings and minimum-weight perfect matchings, the work combines parameterized complexity analysis, combinatorial optimization theory, and bilevel modeling to show that even with a single mandatory or forbidden edge, the follower response problem for general matchings remains NP-hard. In contrast, when every edge is either mandatory or forbidden, the perfect matching variant becomes polynomial-time solvable. Furthermore, the problem is shown to be fixed-parameter tractable with respect to the number of non-mandatory edges, thereby precisely delineating the complexity boundary between these two matching settings under structural constraints.

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.