This work introduces an agentic framework that reformulates a constraint model from an open-ended space and establishes correctness empirically rather than by construction, and demonstrates that autonomous agentic methods can support the improvement of constraint models.
Abstract
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.
Combinatorial problems appear in numerous industrial applications. A common approach is to formulate these problems as declarative constraint models that can subsequently be compiled to and solved by a range of back-end solvers. Recent work shows that Large Language Models (LLMs) can produce correct models from natural language, but even a correct model can be expensive to solve because performance remains sensitive to modelling choices. In this work, we investigate whether LLMs can automate performance-oriented model reformulation. Inspired by Automatic Heuristic Design (AHD), we use an evolutionary framework in which an LLM proposes candidate reformulations that are verified and benchmarked against the user-defined baseline model. We compare AHD-adapted search strategies that control which prior attempts, instructions, and measured feedback enter each prompt. Existing retention strategies prioritize recency or performance, but do not explicitly diversify the context. To cover this gap, we introduce Profile-Diverse Retention (PDR), which applies Maximal Marginal Relevance (MMR) to instance-level runtime vectors to retain behaviourally diverse attempts. We systematically evaluate the strategies on eight CSPLib problems using validation-based final model selection. The results show that: (i) iterative reformulation can produce substantial held-out speedups; (ii) strategies that keep the retained context diverse outperform those that retain only recent or the fastest attempts; and (iii) validation-based selection improves the held-out speedup of every strategy.
Kostis Michailidis, Dimos Tsouros, Nguyen Dang et al.· 0 citations
Background: In declarative constraint solving, a user formulates a constraint model in terms of variables and constraints, and uses a generic, off-the-shelf solver to solve the problem. Constraint Programming (CP) is one such constraint-solving paradigm, which has a wide support for many types of constraints, including global constraints. Global constraints capture complex relations between several decision variables, and CP solvers have specialized propagators to solve them efficiently.
Objectives: While CP solvers support a wide range of global constraints, few solvers support them in a reified or even halfreified context. Hence, for most CP solvers, reified global constraints must be decomposed into non-global constraints. This bypasses the global constraint propagator, which can considerably slow the solving process, suppressing one of the key benefits of modeling with global constraints: a faster solve-time. This is a problem for generic CP-modeling, as reified constraints are often introduced by the modeling system during translation and flattening of compound constraints. Additionally, half-reified global constraints are also used extensively in eXplainable Constraint Programming techniques (XCP). Therefore, XCP techniques suffer from scalability issues when global constraints occur in the model. In this paper, we aim to alleviate this bottleneck, allowing the use of the propagators of global constraints when modeling their half-reification, for all CP-solvers.
Methods: We propose a set of reformulation rules that allow the use of half-reification of a global constraint with any CP solver that supports the “normal” global constraint propagator. This is achieved by introducing auxiliary variables for the decision variables in the global constraint and the use of a reified channeling constraint, which most CP solvers support. Additionally, we show how to reduce the overhead of these auxiliary variables, by limiting the number of variables introduced, and by fixing their value when unconstrained. Finally, we prove the propagation strength of our reformulation for a variety of global constraint families.
Results: We experimentally evaluate the reformulations on a variety of global constraints and applications where reified constraints can occur. Our results show that this reformulation is much faster compared to decomposing the global constraint, and in some cases, even outperforms solver-level propagation routines for reified global constraints.
Conclusions: Using the reformulations proposed in this paper, we make the use of reified global constraints easily available for any CP solver. Hence, we expand the range of available solvers and constraint models that can be used in XCP techniques or for solving CSPs with compound constraints.
Ignace Bleukx, Hélène Verhaeghe, Dimos Tsouros et al.· Journal of Artificial Intell...· 1 citation
This work presents a pipeline that combines LLM-based constraint generation with empirical evaluation and formal verification, and handles MiniZinc’s partial semantics by requiring the base model to be safe and separately proving that the proposed constraint is well-defined for all instances and solutions of the base model.
Philipp Danzinger, Nysret Musliu· International Conference on...· 0 citations
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.
Shaofeng Zhang, Hongyuan Su, Qing Peng et al.· 0 citations
Large language models (LLMs) can translate natural-language optimization problems into solver-ready formulations, but direct code generation is brittle: schema, indexing, and semantic errors can cause compilation failures, infeasible models, or incorrect objectives, while iterative repair, search, and multi-agent workflows increase inference cost. We present IR2Solve, an intermediate-representation-first autoformulation pipeline that uses a single semantic LLM call to produce a schema-constrained ModelIR, followed by two deterministic stages: verification and IR-to-solver compilation. ModelIR explicitly represents sets, parameters, variables, objectives, and constraints using restricted Python-like expression strings. A concrete scalar-constraint convention represents finite per-index constraint families as individual entries, reducing free-index and implicit-quantification errors while simplifying downstream verification and compilation. Across six cleaned optimization benchmarks, IR2Solve achieves strong objective correctness and remains competitive with recent optimization-modeling systems. A controlled ablation on 153 IndustryOR and ComplexLP instances shows sequential gains from the structured IR interface, the scalar-constraint instruction, and deterministic verification. On a matched ten-instance cost panel, IR2Solve uses one semantic call per instance, whereas Chain-of-Experts and SAC-Opt use 8 and 39 calls per instance and consume 3.3 and 22.9 times the token volume of IR2Solve, respectively. These results show that structured intermediate representations, combined with deterministic post-generation processing, provide a practical accuracy-cost trade-off for LLM-based optimization autoformulation.
Penglin Zhu, Linhai Zhang, Jungang Xu et al.· 0 citations
In this work, we focus on SE-RRMs, a symbol-equivariant instantiation of RRMs that exhibits improved extrapolation to larger problem sizes. We propose a neuro-symbolic approach, ``Guiding with Recurrent Reasoning Models''(G-RRM), which integrates SE-RRMs with symbolic solvers for constraint satisfaction problems. SE-RRMs act as neural solvers that generate full solution proposals and guide classical symbolic solvers, such as backtracking or SAT-based methods like Glucose 4.1 and CaDiCaL 3.0.0, that produce globally correct solutions. Centrally, we investigate when neural guidance with G-RRM improves the search efficiency of symbolic solvers. % Our experiments show that the efficacy of G-RRM depends on two conditions: first, the problem instances must have an expansive combinatorial search space to expose potential gains, and second, the solver architecture must be capable of dynamically overwriting its branching choices to recover when neural hints are imperfect. When these conditions hold, guidance drives median conflict counts to zero and yields significant wall-clock speedups: on $9\times9$ Sudoku, where the SE-RRM correctly solves $91.1\%$ of instances, backtracking accelerates by $33.3\times$ and Glucose 4.1 by $1.70\times$ (median, $p<0.001$), with Glucose 4.1 retaining a $1.17\times$ speedup on perfect-hint $25\times25$ grids. In contrast, CaDiCaL 3.0.0, whose runtime is overhead-dominated and which always respects the injected branching hints rather than overwriting them, shows no significant speedup (median $1.02\times$, n.s.) and even a small significant mean slowdown ($0.90\times$) on $9\times9$. These results delineate the regimes in which neural guidance translates into practical speedups.
Timo Bertram, Sidhant Bhavnani, Richard Freinschlag et al.· 0 citations