EquivaMap: Leveraging LLMs for Automatic Equivalence Checking of Optimization Formulations

2025-02-20

AI Summary

Automated semantic equivalence verification among mathematical programming formulations (e.g., ILP/LP) remains a critical challenge in combinatorial optimization modeling—especially amid increasing use of optimization copilots that generate diverse reformulations. Existing approaches rely on ad hoc heuristics lacking formal foundations and verifiability. Method: We propose (1) a formal criterion for *quasi-Karp equivalence*, grounded in variable mappings; (2) the first open-source benchmark dataset of equivalent optimization formulations; and (3) an LLM-driven, interpretable mapping discovery framework integrating prompt engineering, optimization transformation rules (e.g., LP relaxation, valid inequality addition), and a logic-based verification engine. Results: Evaluated on our benchmark, our method significantly outperforms prior techniques in equivalence identification accuracy. It is the first to enable *verifiable*, *traceable*, and *scalable* automated equivalence checking—establishing a rigorous, reproducible foundation for formulation analysis in optimization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Feature Construction/Reformulation

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

A fundamental problem in combinatorial optimization is identifying equivalent formulations, which can lead to more efficient solution strategies and deeper insights into a problem's computational complexity. The need to automatically identify equivalence between problem formulations has grown as optimization copilots--systems that generate problem formulations from natural language descriptions--have proliferated. However, existing approaches to checking formulation equivalence lack grounding, relying on simple heuristics which are insufficient for rigorous validation. Inspired by Karp reductions, in this work we introduce quasi-Karp equivalence, a formal criterion for determining when two optimization formulations are equivalent based on the existence of a mapping between their decision variables. We propose EquivaMap, a framework that leverages large language models to automatically discover such mappings, enabling scalable and reliable equivalence verification. To evaluate our approach, we construct the first open-source dataset of equivalent optimization formulations, generated by applying transformations such as adding slack variables or valid inequalities to existing formulations. Empirically, EquivaMap significantly outperforms existing methods, achieving substantial improvements in correctly identifying formulation equivalence.

Problem

• Automated equivalence checking of optimization formulations
• Scalable verification using large language models
• Improving efficiency in combinatorial optimization strategies

Innovation

• Leverages LLMs for equivalence checking
• Introduces quasi-Karp equivalence criterion
• Creates open-source dataset for validation