Practical Polytope Volume Approximation

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear halfspaces. We implement and evaluate randomized polynomial-time algorithms for accurately approximating the polytope’s volume in high dimensions (e.g., few hundreds) based onhit-and-run random walks. To carry out this efficiently, we experimentally correlate the effect of parameters, such as random walk length and number of sample points, with accuracy and runtime. Our method is based on Monte Carlo algorithms with guaranteed speed and provably high probability of success for arbitrarily high precision. We exploit the problem’s features in implementing a practical rounding procedure of polytopes, in computing only partial “generations” of random points, and in designing fast polytope boundary oracles. Our publicly available software is significantly faster than exact computation and more accurate than existing approximation methods. For illustration, volume approximations of Birkhoff polytopes B₁₁,…,B₁₅ are computed, in dimensions up to 196, whereas exact methods have only computed volumes of up to B₁₀.