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In this paper we consider a simple variant of the Online Dial-a-Ride Problem from a probabilistic point of view. To this end, we look at a probabilistic version of this online Dial-a-Ride problem and introduce a probabilistic notion of the competitive ratio which states that an algorithm performs well on the vast majority of the instances. Our main result is that under the assumption of high load a certain online algorithm is probabilistically $(1+o(1))$-competitive if the underlying graph is a tree. This result can be extended to general graphs by using well-known approximation techniques at the expense of a distortion factor~$O(\log\|V\|)$.
We study combinatorial structures in large-scale mixed-integer (nonlinear) programming problems arising in gas network optimization. We propose a preprocessing strategy exploiting the observation that a large part of the combinatorial complexity arises in certain subnetworks. Our approach analyzes these subnetworks and the combinatorial structure of the flows within these subnetworks in order to provide alternative models with a stronger combinatorial structure that can be exploited by off-the-shelve solvers. In particular, we consider the modeling of operation modes for complex compressor stations (i.e., ones with several in- or outlets) in gas networks. We propose a refined model that allows to precompute tighter bounds for each operation mode and a number of model variants based on the refined model exploiting these tighter bounds. We provide a procedure to obtain the refined model from the input data for the original model. This procedure is based on a nontrivial reduction of the graph representing the gas flow through the compressor station in an operation mode. We evaluate our model variants on reference benchmark data, showing that they reduce the average running time between 10% for easy instances and 46% for hard instances. Moreover, for three of four considered networks, the average number of search tree nodes is at least halved, showing the effectivity of our model variants to guide the solver’s search.
In \emph{classical optimization} it is assumed that full information about the problem to be solved is given. This, in particular, includes that all data are at hand. The real world may not be so nice'' to optimizers. Some problem constraints may not be known, the data may be corrupted, or some data may not be available at the moments when decisions have to be made. The last issue is the subject of \emph{online optimization} which will be addressed here. We explain some theory that has been developed to cope with such situations and provide examples from practice where unavailable information is not the result of bad data handling but an inevitable phenomenon.
In a recent paper, Conte et al. [CGMR2017] presented an algorithm for enumerating all acyclic orientations of a graph G=(V,E) with a single source (and related orientations) with delay O(|V||E|). In this paper we revisit the problem by going back to an early paper by de Fraysseix et al. [FMR1995], who proposed an algorithm for enumerating all bipolar orientations of a graph based on a recursion formula. We first formalize de Fraysseix et al.'s algorithm for bipolar orientations and determine that its delay is also O(|V||E|). We then apply their recursion formula to the case of Conte et al.'s enumeration problem and show that this yields a more efficient enumeration algorithm with delay O(\sqrt(|V|)|E|). Finally, a way to further streamline the algorithm that leads to a particularly simple implementation is suggested.