Abstract : We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytope's volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problem's geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B 11 ,. .. , B 15 , whereas exact methods have only computed that of B 10 .
https://hal.inria.fr/hal-01897272
Contributor : Ioannis Emiris
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Submitted on : Sunday, October 28, 2018 - 12:43:28 PM
Last modification on : Monday, October 29, 2018 - 9:52:11 AM
Long-term archiving on : Tuesday, January 29, 2019 - 12:50:14 PM
Ioannis Z. Emiris, Vissarion Fisikopoulos. Efficient Random-Walk Methods for Approximating Polytope Volume. ACM Transactions on Mathematical Software, Association for Computing Machinery, 2018, 44 (4), pp.1 - 21. ⟨10.1145/3194656⟩. ⟨hal-01897272⟩
