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A b s t r ac t . The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten "rules of thumb". I n t ro d u c t i o nIn integer and linear optimization the software workhorses are solvers for linear programs (based on simplex or interior point methods) as well as generic frameworks for branch-andbound or branch-and-cut schemes. Comprehensive implementations are available both as Open Source, like SCIP [2], as well as commercial software, like CPLEX [24] and Gurobi [39]. While today it is common to solve linear programs with millions of rows and columns and, moreover, mixed integer linear programs with sometimes hundreds of thousands of rows and columns, big challenges remain. For instance, the 2010 version of the MIPLIB [30] lists the mixed-integer problem liu with 2 178 rows, 1 156 columns, and a total of only 10 626 non-zero coefficients; this seems to be impossible to solve with current techniques. One way to make progress in the field is to invent new families of cutting planes, either of a general kind or specifically tailored to a class of examples. In the latter situation the strongest possible cuts are obviously those arising from the facets of the (mixed) integer hull. A main purpose of this note is to report on the state of the art of getting at such facets in a brute force kind of way. And we will do so by explaining how our software system polymake [56] can help.Here we focus on integer linear programming (ILP); mixed integer linear programming (MILP) will only be mentioned in passing. To avoid technical ramifications we assume that all our linear programs (LP) are bounded. The brute force method for obtaining all facets of the integer hull is plain and simple, and it has two steps. First, we compute all the feasible integer points. Since we assumed boundedness these are only finitely many. Second, we compute the facets of their convex hull. Of course, the catch is that neither problem is really easy. Deciding if an ILP has an (integer) feasible point is known to be NP-complete [35]. So, we may not even hope for any efficient algorithm for the first step. Most likely, the situation for the second is about equally bad. While it is open whether or not there is a convex hull algorithm which runs in polynomial time measured in the combined sizes of the input and the output, recent work of Khachiyan et al.[47] indicates a negative answer. They show that computing the vertices of an unbounded polyhedron is hard; the difference to the general convex hull problem is that their result does not say anything about the rays of the polyhedron.Our paper is organized as follows. We start out with a very brief introduction to the polymake system and its usage. In Section 3 we explore how various convex hull algorithms and their implementations behave on various kinds of input. Our inpu...

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.

We present an algorithm for the problem of routing Automated Guided Vehicles (AGVs) in an automated logistic system. The algorithm avoids collisions, deadlocks and livelocks already at the time of route computation (conflict-free routing). After a preprocessing step the real-time computation for each request consists of the determination of a shortest path with time-windows and a following readjustment of these time-windows. Both is done in polynomial-time. Using goal-oriented search we get computation times which are appropriate for real-time routing. Additionally, in comparison to a static routing approach, used in Container Terminal Altenwerder (CTA) at Hamburg Harbour, our algorithm had an explicit advantage.

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