Approximating the volume of tropical polytopes is difficult

Abstract

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 α = 2 poly(m,n) for the volume of a tropical polytope given by n vertices in a space of dimension m, unless P=NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. If 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. For tropical polytopes described by inequalities we prove that counting the number of integer points and calculating the volume are #P-hard.

Tropical Ehrhart theory and tropical volume

“…Going back to tropical canonical lattice polytopes (see Definition 2.6), we actually obtain two different polynomials; one counting the lattice points in Z d , the other one counting b-lattice points. The first version is less natural from the semiring operations, but it was used in [22]. The next concept is at the heart of our quantitative studies.…”

“…In particular, as lattices correspond to discrete additive subgroups of , they have a fix-group of translations. Although this perspective has been used in [ 22 ] and allows a tropical Ehrhart theory connected to the Euclidean volume of the polytopes in the covector decomposition (see Sect. 3 ), it is too rough for our purposes.…”

“…Measuring quantities from tropical geometry turned out to be fruitful for a better understanding of interior point methods for linear programming [ 2 ] and principal component analysis of biological data [ 37 ]. Moreover, it has interesting connections with representation theory [ 29 , 38 ] and computational complexity [ 22 ].…”

“…Former work only considered the lattice points in a d -dimensional polytope, see in particular [ 12 ]. This idea was used to measure its Euclidean volume and deduce the hardness to compute it by counting the integer lattice points [ 22 ]. These lattice points arise naturally through the representation of affine buildings as tropical polytopes [ 29 ].…”

“…Therefore, this decision problem lies in NP coNP (cf. [ 22 ]). This equivalence is analogous to the classical setting, where existence of interior points in a polytope is equivalent to solving linear programs (cf.…”