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The Generalized Baues Problem asks whether for a given point configuration the order complex of all its proper polyhedral subdivisions, partially ordered by refinement, is homotopy equivalent to a sphere. In this paper, an affirmative answer is given for the vertex sets of cyclic polytopes in all dimensions. This yields the first non-trivial class of point configurations with neither a bound on the dimension, the codimension, nor the number of vertice for which this is known to be true. Moreover, it is shown that all triangulations of cyclic polytopes are lifting triangulations. This contrasts the fact that in general there are many non-regular triangulations of cyclic polytopes. Beyond this, we find triangulations of $C(11,5)$ with flip deficiency. This proves---among other things---that there are triangulations of cyclic polytopes that are non-regular for every choice of points on the moment curve.
In this paper we consider the following online transportation problem (\textsc{Oltp}): Objects are to be transported between the vertices of a given graph. Transportation requests arrive online, specifying the objects to be transported and the corresponding source and target vertex. These requests are to be handled by a server which commences its work at a designated origin vertex and which picks up and drops objects at their starts and destinations. After the end of its service the server returns to its start. The goal of \textsc{Oltp} is to come up with a transportation schedule for the server which finishes as early as possible. We first show a lower bound of~$5/3$ for the competitive ratio of any deterministic algorithm. We then analyze two simple and natural strategies which we call \textsf{REPLAN} and \textsf{IGNORE}. \textsf{REPLAN} completely discards its schedule and recomputes a new one when a new request arrives. \textsf{IGNORE} always runs a (locally optimal) schedule for a set of known requests and ignores all new requests until this schedule is completed. We show that both strategies, \textsf{REPLAN} and \textsf{IGNORE}, are $5/2$-competitive. We also present a somewhat less natural strategy \textsf{SLEEP}, which in contrast to the other two strategies may leave the server idle from time to time although unserved requests are known. We also establish a competitive ratio of~$5/2$ for the algorithm \textsf{SLEEP}. Our results are extended to the case of ``open schedules'' where the server is not required to return to its start position at the end of its service.
In ``classical'' optimization, all data of a problem instance are considered given. The standard theory and the usual algorithmic techniques apply to such cases only. Online optimization is different. Many decisions have to be made before all data are available. In addition, decisions once made cannot be changed. How should one act ``best'' in such an environment? In this paper we survey online problems coming up in combinatorial optimization. We first outline theoretical concepts, such as competitiveness against various adversaries, to analyze online problems and algorithms. The focus, however, lies on real-world applications. We report, in particular, on theoretical investigations and our practical experience with problems arising in transportation and the automatic handling of material.
This paper gives a short introduction into combinatorial online optimization. It explains a few evaluation concepts of online algorithms, such as competitiveness, and discusses limitations in their application to real--world problems. The main focus, however, is a survey of combinatorial online problems coming up in practice, in particular, in large scale material flow and flexible manufacturing systems.