We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes. Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).

We specify what is meant for a polytope to be reconstructible from its graph or dual graph, and we introduce the problem of class reconstructibility; i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present an $O(n^3)$ algorithm that computes the vertices of a matroid polytope from its $n$-vertex graph. Moreover, our proof includes a characterization of all matroids with isomorphic basis exchange graphs.

Wireless sensor networks (WSNs) are often advertised with high sensing accuracy, long lifetime, and easy deployment. However, they are still not widely used in environmental research due to of poor tool support and high complexity. A wider use of WSNs in field science would enable researchers to address scientific questions that are infeasible today.