Geometric Reasoning with polymake

The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background

simpcomp -- A GAP toolbox for simplicial complexes

simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of manifolds.Comment: 4 page

Conflict-free Real-time AGV Routing

In automated logistic systems Automated Guided Vehicles (AGVs) are used for transportation tasks. To deal with the interaction in such an AGV system one needs efficient and intelligent routing on the one hand and collision avoidance on the other. Obviously, AGV routing is an online routing problem (nothing is known about future requests) and even a real-time problem, because fast answers are required. A route should be computed in less than a second

Dynamic Routing of Automated Guided Vehicles in Real-time

Automated Guided Vehicles (AGVs) are state-of-the-art technology for optimizing large scale production systems and are used in a wide range of application areas. A standard task in this context is to find efficient routing schemes, i.e., algorithms that route these vehicles through the particular environment. The productivity of the AGVs is highly dependent on the used routing scheme. In this work we study a particular routing algorithm for AGVs in an automated logistic system. For the evaluation of our algorithm we focus on Container Terminal Altenwerder~(CTA) at Hamburg Harbor. However, our model is appropriate for an arbitrary graph. The key feature of this algorithm is that it avoids collisions, deadlocks and livelocks already at the time of route computation (conflict-free routing), whereas standard approaches deal with these problems only at the execution time of the routes. In addition, the algorithm considers physical properties of the AGVs and certain safety aspects implied by the particular application

Extended formulations for column constrained orbitopes

In the literature, packing and partitioning orbitopes were discussed to handle symmetries that act on variable matrices in certain binary programs. In this paper, we extend this concept by restrictions on the number of $1$-entries in each column. We develop extended formulations of the resulting polytopes and present numerical results that show their effect on the LP relaxation of a graph partitioning problem

The finiteness threshold width of lattice polytopes

In each dimension d there is a constant woo(d) [épsilon] N such that for every n [épsilon] N all but finitely many lattice d-polytopes with n lattice points have llattice width at most woo(d). We call woo(d) the finiteness threshold width in dimension d and show that d - 2 woo(d) O _d4/3_. Blanco and Santos determined the value woo(3) = 1. Here, we establish woo(4) = 2. This implies, in particular, that there are only finitely many empty 4-simplices of width larger than two. (This last statement was claimed by Barile et al. in [Proc. Am. Math. Soc. 139 (2011), pp. 4247-4253], but we have found a gap in their proof.) Our main tool is the study of d-dimensional lifts of hollow (d-1)-polytopes.The first and fourth authors were supported by grants MTM2014-54207-P, MTM2017-83750-P, and the first author was also supported by BES-2012-058920 of the Spanish Ministry of Science. The fourth author was also supported by the Einstein Foundation Berlin under grant EVF-2015-230. The third author was supported by the Berlin Mathematical School