Efficient Compilation of Algorithms into Compact Linear Programs

Abstract: Linear Programming (LP) is widely applied in industry and is a key component of various other mathematical problem-solving techniques. Recent work introduced an LP compiler translating polynomial-time, polynomial-space algorithms into polynomial-size LPs using intuitive high-level programming languages, offering a promising alternative to manually specifying each set of constraints through Algebraic Modeling Languages (AMLs). However, the resulting LPs, while polynomial in size, are often extremely large, posing challenges for existing LP solvers. In this paper, we propose a novel approach for generating substantially smaller LPs from algorithms. Our goal is to establish minimum-size compact LP formulations for problems in P having natural formulations with exponential extension complexities. Our broader vision is to enable the systematic generation of Compact Integer Programming (CIP) formulations for problems with exponential-size IPs having polynomial-time separation oracles. To this end, we introduce a hierarchical linear pipelining technique that decomposes nested program structures into synchronized regions with well-defined execution transitions -- functions of compile-time parameters. This decomposition allows us to localize LP constraints and variables within each region, significantly reducing LP size without the loss of generality, ensuring the resulting LP remains valid for all inputs of size $n$. We demonstrate the effectiveness of our method on two benchmark problems -- the makespan problem, which has exponential extension complexity, and the weighted minimum spanning tree problem -- both of which have exponential-size natural LPs. Our results show up to a $25$-fold reduction in LP size and substantial improvements in solver performance across both commercial and non-commercial LP solvers.