Approximating the volume of tropical polytopes is difficult
Stéphane Gaubert, Marie Maccaig · International Journal of Algebra and Computation · 2018
We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor [Formula: see text] for the volume of a tropical polytope given by [Formula: see text] for the volume of a tropical polytope given by [Formula: see text] vertices in a space of dimension [Formula: see text], unless P[Formula: see text]NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. It follows that approximating these values for tropical polytopes is more difficult than for classical polytopes. Our proofs use a reduction from the problem of calculating the tropical rank.