## Report in Wirtschaftsmathematik (WIMA Report)

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- (dynamic) network flows (1)
- Bell Number (1)
- Berechnungskomplexität (1)
- Competitive Analysis (1)
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- Dynamic Network Flows (1)
- FPTAS (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.

- 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.

- 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.

- 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.

- 116
- How to find Nash equilibria with extreme total latency in network congestion games? (2008)
- We study the complexity of finding extreme pure Nash equilibria in symmetric network congestion games and analyse how it depends on the graph topology and the number of users. In our context best and worst equilibria are those with minimum respectively maximum total latency. We establish that both problems can be solved by a Greedy algorithm with a suitable tie breaking rule on parallel links. On series-parallel graphs finding a worst Nash equilibrium is NP-hard for two or more users while finding a best one is solvable in polynomial time for two users and NP-hard for three or more. Additionally we establish NP-hardness in the strong sense for the problem of finding a worst Nash equilibrium on a general acyclic graph.

- 86
- Locating stops along bus or railway lines - a bicriterial problem (2003)
- In this paper we consider the location of stops along the edges of an already existing public transportation network, as introduced in [SHLW02]. This can be the introduction of bus stops along some given bus routes, or of railway stations along the tracks in a railway network. The goal is to achieve a maximal covering of given demand points with a minimal number of stops. This bicriterial problem is in general NP-hard. We present a nite dominating set yielding an IP-formulation as a bicriterial set covering problem. We use this formulation to observe that along one single straight line the bicriterial stop location problem can be solved in polynomial time and present an e cient solution approach for this case. It can be used as the basis of an algorithm tackling real-world instances.

- 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.

- 117
- Minimum Cut Tree Games (2008)
- In this paper we introduce a cooperative game based on the minimum cut tree problem which is also known as multi-terminal maximum flow problem. Minimum cut tree games are shown to be totally balanced and a solution in their core can be obtained in polynomial time. This special core allocation is closely related to the solution of the original graph theoretical problem. We give an example showing that the game is not supermodular in general, however, it is for special cases and for some of those we give an explicit formula for the calculation of the Shapley value.

- 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.