Abstract:The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML (extensible markup language) format. In this paper, we explain this format and present the instances that are available in the library.Data Set: http://gaslib.zib.de Data Set License: CC BY 3.0 Keywords: gas transport; networks; problem instances; mixed-integer nonlinear optimization; GasLib MSC: 90-08; 90C90; 90B10
SummaryThe mathematical simulation and optimization of gas transport through pipeline systems is an important field of research with a large practical impact. Over the last decades, many different mathematical models on different levels of accuracy for different components of gas networks have been developed. On the basis of these models, several simulation and optimization algorithms have been proposed. We refer to [1][2][3] and the references therein for more information. With GasLib, we provide a set of network instances that can be used to test and compare such models and the algorithms for solving them.
Abstract. We show that the edge graph of a 6-dimensional polytope with 12 facets has diameter at most 6, thus verifying the d-step conjecture of Klee and Walkup in the case of d = 6. This implies that for all pairs (d, n) with n−d ≤ 6 the diameter of the edge graph of a d-polytope with n facets is bounded by 6, which proves the Hirsch conjecture for all n − d ≤ 6. We show this result by showing this bound for a more general structure -so-called matroid polytopes -by reduction to a small number of satisfiability problems. We show that the d-step conjecture is true in dimension 6 as well. We derive this result by considering a more general class of objects, namely matroid polytopes, i.e. oriented matroids, which, if realizable, correspond to convex polytopes. We show that no 6-dimensional matroid polytope with 12 vertices and a facet path of length 7 exists. Then ∆(6, 12) = 6 follows by considering polarity and the already known bounds.To show that ∆(6, 12) ≤ 6 we first give combinatorial conditions for matroid polytopes that violate this bound. This achieved through the study of path complexes (see Bremner et al. [2], cf. Section 1). We then show that these conditions cannot be satisfied by an oriented matroid. To show this we use a satisfiability solver to produce the desired contradiction (see Section 2). We will use the same method to show that ∆(4, 11) = 6, which settles another special case of the Hirsch conjecture. The latter result allows us to also improve the upper bound on ∆(5, 12) from 9 to 8.For small parameters there are already known general bounds that allow us to compute or at least bound the diameter of polytopes. We summarize them in the following Lemma. For an overview about the known bounds we refer to the respective chapters in the books by Grünbaum [6] (1) ∆(3, n) =
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.