Sparktope: linear programs from algorithms

Abstract

In a recent paper, Avis, Bremner, Tiwary and Watanabe gave a method for constructing linear programs (LPs) based on algorithms written in a simple programming language called Sparks. If an algorithm produces the solution x to a problem in polynomial time and space then the LP constructed is also of polynomial size and its optimum solution contains x as well as a complete execution trace of the algorithm. Their method led us to the construction of a compiler called sparktope which we describe in this paper. This compiler allows one to generate polynomial sized LPs for problems in that have exponential extension complexity, such as matching problems in non-bipartite graphs. In this paper, we describe sparktope, the language Sparks, and the assembler instructions and LP constraints it produces. This is followed by two concrete examples, the makespan problem and the problem of testing if a matching in a graph is maximum, both of which are known to have exponential extension complexity. Computational results are given. In discussing these examples, we make use of visualization techniques included in sparktope that may be of independent interest. The extremely large linear programs produced by the compiler appear to be quite challenging to solve using currently available software. Since the optimum LP solutions can be computed independently they may be useful as benchmarks. Further enhancements of the compiler and its application are also discussed.