## Report in Wirtschaftsmathematik (WIMA Report)

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- (dynamic) network flows (1)
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- 142
- An online approach to detecting changes in nonlinear autoregressive models (2011)
- In this paper we develop monitoring schemes for detecting structural changes in nonlinear autoregressive models. We approximate the regression function by a single layer feedforward neural network. We show that CUSUM-type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression in both an offline as well as online setting, can be extended to this model. The proposed monitoring schemes reject (asymptotically) the null hypothesis only with a given probability but will detect a large class of alternatives with probability one. In order to construct these sequential size tests the limit distribution under the null hypothesis is obtained.

- 136
- The Multi Terminal q-FlowLoc Problem: A Heuristic (2011)
- In this paper the multi terminal q-FlowLoc problem (q-MT-FlowLoc) is introduced. FlowLoc problems combine two well-known modeling tools: (dynamic) network flows and locational analysis. Since the q-MT-FlowLoc problem is NP-hard we give a mixed integer programming formulation and propose a heuristic which obtains a feasible solution by calculating a maximum flow in a special graph H. If this flow is also a minimum cost flow, various versions of the heuristic can be obtained by the use of different cost functions. The quality of this solutions is compared.

- 135
- Reliable and Restricted Quickest Path Problems (2011)
- In a dynamic network, the quickest path problem asks for a path minimizing the time needed to send a given amount of flow from source to sink along this path. In practical settings, for example in evacuation or transportation planning, the reliability of network arcs depends on the specific scenario of interest. In this circumstance, the question of finding a quickest path among all those having at least a desired path reliability arises. In this article, this reliable quickest path problem is solved by transforming it to the restricted quickest path problem. In the latter, each arc is associated a nonnegative cost value and the goal is to find a quickest path among those not exceeding a predefined budget with respect to the overall (additive) cost value. For both, the restricted and reliable quickest path problem, pseudopolynomial exact algorithms and fully polynomial-time approximation schemes are proposed.

- 132
- Train Marshalling Problem - Algorithms and Bounds - (2010)
- The Train Marshalling Problem consists of rearranging an incoming train in a marshalling yard in such a way that cars with the same destinations appear consecutively in the final train and the number of needed sorting tracks is minimized. Besides an initial roll-in operation, just one pull-out operation is allowed. This problem was introduced by Dahlhaus et al. who also showed that the problem is NP-complete. In this paper, we provide a new lower bound on the optimal objective value by partitioning an appropriate interval graph. Furthermore, we consider the corresponding online problem, for which we provide upper and lower bounds on the competitiveness and a corresponding optimal deterministic online algorithm. We provide an experimental evaluation of our lower bound and algorithm which shows the practical tightness of the results.

- 130
- Min-Max Quickest Path Problems (2010)
- In a dynamic network, the quickest path problem asks for a path such that a given amount of flow can be sent from source to sink via this path in minimal time. In practical settings, for example in evacuation or transportation planning, the problem parameters might not be known exactly a-priori. It is therefore of interest to consider robust versions of these problems in which travel times and/or capacities of arcs depend on a certain scenario. In this article, min-max versions of robust quickest path problems are investigated and, depending on their complexity status, exact algorithms or fully polynomial-time approximation schemes are proposed.

- 129
- Dynamic Multi-Period Routing With Two Classes (2010)
- In the Dynamic Multi-Period Routing Problem, one is given a new set of requests at the beginning of each time period. The aim is to assign requests to dates such that all requests are fulfilled by their deadline and such that the total cost for fulling the requests is minimized. We consider a generalization of the problem which allows two classes of requests: The 1st class requests can only be fulfilled by the 1st class server, whereas the 2nd class requests can be fulfilled by either the 1st or 2nd class server. For each tour, the 1st class server incurs a cost that is alpha times the cost of the 2nd class server, and in each period, only one server can be used. At the beginning of each period, the new requests need to be assigned to service dates. The aim is to make these assignments such that the sum of the costs for all tours over the planning horizon is minimized. We study the problem with requests located on the nonnegative real line and prove that there cannot be a deterministic online algorithm with a competitive ratio better than alpha. However, if we require the difference between release and deadline date to be equal for all requests, we can show that there is a min{2*alpha, 2 + 2/alpha}-competitive algorithm.

- 127
- Online Delay Management (2010)
- We present extensions to the Online Delay Management Problem on a Single Train Line. While a train travels along the line, it learns at each station how many of the passengers wanting to board the train have a delay of delta. If the train does not wait for them, they get delayed even more since they have to wait for the next train. Otherwise, the train waits and those passengers who were on time are delayed by delta. The problem consists in deciding when to wait in order to minimize the total delay of all passengers on the train line. We provide an improved lower bound on the competitive ratio of any deterministic online algorithm solving the problem using game tree evaluation. For the extension of the original model to two possible passenger delays delta_1 and delta_2, we present a 3-competitive deterministic online algorithm. Moreover, we study an objective function modeling the refund system of the German national railway company, which pays passengers with a delay of at least Delta a part of their ticket price back. In this setting, the aim is to maximize the profit. We show that there cannot be a deterministic competitive online algorithm for this problem and present a 2-competitive randomized algorithm.

- 125
- Generalized Max Flow in Series-Parallel Graphs (2010)
- In the generalized max flow problem, the aim is to find a maximum flow in a generalized network, i.e., a network with multipliers on the arcs that specify which portion of the flow entering an arc at its tail node reaches its head node. We consider this problem for the class of series-parallel graphs. First, we study the continuous case of the problem and prove that it can be solved using a greedy approach. Based on this result, we present a combinatorial algorithm that runs in O(m*m) time and a dynamic programming algorithm with running time O(m*log(m)) that only computes the maximum flow value but not the flow itself. For the integral version of the problem, which is known to be NP-complete, we present a pseudo-polynomial algorithm.

- 123
- Selfish Bin Coloring (2009)
- We introduce a new game, the so-called bin coloring game, in which selfish players control colored items and each player aims at packing its item into a bin with as few different colors as possible. We establish the existence of Nash and strong as well as weakly and strictly Pareto optimal equilibria in these games in the cases of capacitated and uncapacitated bins. For both kinds of games we determine the prices of anarchy and stability concerning those four equilibrium concepts. Furthermore, we show that extreme Nash equilibria, those with minimal or maximal number of colors in a bin, can be found in time polynomial in the number of items for the uncapcitated case.

- 122
- Earliest Arrival Flows in Series-Parallel Graphs (2009)
- We present an exact algorithm for computing an earliest arrival flow in a discrete time setting on series-parallel graphs. In contrast to previous results for the earliest arrival flow problem this algorithm runs in polynomial time.