design ilp formulations

This paper investigates the parameterized complexity of Integer Linear Programming (ILP) with lower and upper bounds, parameterized by the “distance to generalized matching”—the minimum number of modifications required to transform the constraint matrix into one where each column has ℓ₁-norm at most two (encompassing polynomial-time solvable matching and flow problems). The authors introduce two novel structural parameters: a *variable backdoor* (minimum column deletions to achieve the generalized matching structure) and a *constraint backdoor* (minimum row deletions). They present the first fixed-parameter tractable (FPT) algorithm parameterized by variable backdoor size (p). In contrast, they prove W[1]-hardness parameterized by constraint backdoor size (h) and devise a randomized XP algorithm for unary-encoded instances. Key technical innovations include a variant of lattice convexity, Graver basis-enhanced local search, and a pseudo-polynomial reduction to Exact Matching—enabling both tight complexity classification and significant algorithmic advances.

This work addresses the efficiency bottleneck in MaxSAT solving caused by frequent calls to ILP solvers. It presents the first systematic investigation into the impact of ILP preprocessing techniques on MaxSAT solving. We propose an ILP constraint reduction and equivalence substitution method specifically tailored for MaxSAT, which structurally simplifies soft and hard constraints in the ILP encoding prior to solving. This significantly reduces the dependency of WMaxCDCL-style solvers on underlying ILP solvers—without compromising solution accuracy or quality—thereby enhancing the robustness and generalizability of solver portfolios. Experimental evaluation on standard benchmarks shows that WMaxCDCL-OpenWbo1200 solves 15 additional instances; ILP solver invocations decrease substantially; and overall throughput and stability improve markedly.

This work addresses key challenges in integer linear programming (ILP) solvers—limited generalization, reliance on external solvers, and poor efficiency in multimodal energy landscapes—by proposing a training-free, solver-free sampling-based optimization framework that directly explores the discrete feasible region. Leveraging the linear structure of ILP, the method designs a proposal distribution satisfying detailed balance and incorporates a dual tempering mechanism that jointly modulates temperature and penalty parameters to dynamically adjust constraint barriers while preserving the original objective function, thereby enhancing global exploration. Experiments demonstrate that the approach consistently outperforms SCIP across four benchmarks, matches or exceeds Gurobi’s performance on two tasks within 200 seconds, exhibits superior robustness on out-of-distribution instances, and competes effectively with classical solvers on MIPLIB 2017 without any parameter tuning.

Predicting size-extensive molecular properties (e.g., energy, polarizability) across scales—particularly for molecules larger than those in the training set—remains a critical challenge in molecular machine learning. Method: We propose an unsupervised training set selection method based on integer linear programming (ILP), the first to formulate molecular subset selection as an ILP problem. Unlike conventional diversity- or coverage-driven strategies, our approach imposes atom-level local environment similarity constraints to ensure systematic coverage of local chemical motifs while guaranteeing globally optimal subset selection. It integrates physics-informed atomic feature embeddings with ILP-based optimization. Results: The selected training sets significantly improve model generalization to unseen, especially out-of-distribution, large molecules. Experiments across multiple extensive property prediction tasks demonstrate substantial gains over state-of-the-art unsupervised selection baselines, with high computational efficiency and inherent theoretical interpretability.

Embedding decision trees—including ensembles—into optimization problems suffers from low modeling accuracy and poor computational efficiency due to weak linear relaxations in existing mixed-integer programming (MIP) formulations. Method: We propose an ideal MIP formulation based on the union of projection polytopes, explicitly capturing tree logic via binary feature representations and extending to one-dimensional continuous features. Contribution/Results: We prove, for the first time under binary feature encoding, that allowing repeated splits on the same feature eliminates fractional extreme points in the linear relaxation. We further derive the ideal MIP characterization for univariate continuous features. Our formulation substantially tightens the linear relaxation and reduces the number of extreme points in the feasible region. On low-dimensional feature instances, average solution time decreases by an order of magnitude. This advancement significantly improves both the embeddability of tree models into optimization frameworks and their computational scalability.

This paper investigates the geometric essence of the Reformulation-Linearization Technique (RLT) in binary mixed-integer optimization, aiming to clarify its strengthening advantage over the dual of disjunctive programming (DP) for single-variable disjunctions. Method: Employing convex geometric analysis, the authors characterize—geometrically for the first time—the points belonging to the RLT closure and rigorously compare its strength against the DP dual relaxation. Contribution/Results: They establish that RLT strictly dominates the DP dual even in the boundary case where the right-hand side of the cardinality equality constraint equals one—thereby extending the known applicability boundary beyond prior literature. Furthermore, they generalize this dominance result to the broader class of cardinality equality constraints, systematically advancing the theoretical understanding of relative strength among linearization-based strengthening methods for problems such as quadratic assignment.

This study addresses the unclear practical efficacy of automatically generated polynomial symmetry-breaking constraints in integer linear programming across different solvers. The authors systematically evaluate the performance of mainstream mathematical programming and SMT solvers when handling such constraints, comparing three strategies: native quadratic handling, internal reformulation, and explicit linearization. Their experiments reveal that the effectiveness of symmetry breaking is highly solver-dependent, advocating for a solver-aware evaluation paradigm. The findings indicate that compact families of quadratic symmetry-breaking constraints generally enhance solver performance, whereas excessive linearization, overly large breaking sets, or inappropriate reformulations often lead to model bloating or search degradation, thereby diminishing or even reversing potential benefits.

This work addresses the computational inefficiency caused by symmetry in integer programming, which leads to redundant search. The authors propose an algebraic approach that leverages arbitrary basis polynomials and problem-specific permutation groups to automatically generate random polynomial inequalities for symmetry breaking. This method constitutes the first general-purpose framework for generating symmetry-breaking constraints grounded in algebraic structure, applicable to any permutation group and seamlessly integrable into mainstream symbolic computation platforms. Experimental results on nearly half-capacity 0–1 bin packing instances demonstrate that incorporating only a small number of quadratic symmetry-breaking constraints significantly reduces solution time, with the most consistent performance gains observed when combining limited variables and permutations.

Traditional mixed-integer linear programming (MILP) modeling struggles to accommodate the diversity of complex combinatorial optimization problems, resulting in cumbersome formulations and limited solver compatibility. This work proposes OptiDSL, a novel domain-specific language (DSL)-centric modeling paradigm that leverages large language models to automatically translate natural language descriptions into structured DSL representations, thereby decoupling modeling from solving. By moving beyond MILP constraints, the framework enables flexible integration of diverse specialized, heuristic, and learning-based solvers. Evaluated across 44 combinatorial optimization problem classes, OptiDSL achieves a 51.66% improvement in modeling accuracy and reduces modeling time by 91.71% compared to conventional MILP approaches, while further boosting solution accuracy by 23.09% over existing benchmarks.