## Report in Wirtschaftsmathematik (WIMA Report)

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- 145
- The Generalized Assignment Problem with Minimum Quantities (2012)
- We consider a variant of the generalized assignment problem (GAP) where the amount of space used in each bin is restricted to be either zero (if the bin is not opened) or above a given lower bound (a minimum quantity). We provide several complexity results for different versions of the problem and give polynomial time exact algorithms and approximation algorithms for restricted cases. For the most general version of the problem, we show that it does not admit a polynomial time approximation algorithm (unless P=NP), even for the case of a single bin. This motivates to study dual approximation algorithms that compute solutions violating the bin capacities and minimum quantities by a constant factor. When the number of bins is fixed and the minimum quantity of each bin is at least a factor \(\delta>1\) larger than the largest size of an item in the bin, we show how to obtain a polynomial time dual approximation algorithm that computes a solution violating the minimum quantities and bin capacities by at most a factor \(1-\frac{1}{\delta}\) and \(1+\frac{1}{\delta}\), respectively, and whose profit is at least as large as the profit of the best solution that satisfies the minimum quantities and bin capacities strictly. In particular, for \(\delta=2\), we obtain a polynomial time (1,2)-approximation algorithm.

- 144
- A limitation of the estimation of intrinsic volumes via pixel configuration counts (2012)
- It is often helpful to compute the intrinsic volumes of a set of which only a pixel image is observed. A computational efficient approach, which is suggested by several authors and used in practice, is to approximate the intrinsic volumes by a linear functional of the pixel configuration histogram. Here we want to examine, whether there is an optimal way of choosing this linear functional, where we will use a quite natural optimality criterion that has already been applied successfully for the estimation of the surface area. We will see that for intrinsic volumes other than volume or surface area this optimality criterion cannot be used, since estimators which ignore the data and return constant values are optimal w.r.t. this criterion. This shows that one has to be very careful, when intrinsic volumes are approximated by a linear functional of the pixel configuration histogram.

- 143
- Complexity and Approximability of the Maximum Flow Problem with Minimum Quantities (2012)
- We consider the maximum flow problem with minimum quantities (MFPMQ), which is a variant of the maximum flow problem where the flow on each arc in the network is restricted to be either zero or above a given lower bound (a minimum quantity), which may depend on the arc. This problem has recently been shown to be weakly NP-complete even on series-parallel graphs. In this paper, we provide further complexity and approximability results for MFPMQ and several special cases. We first show that it is strongly NP-hard to approximate MFPMQ on general graphs (and even bipartite graphs) within any positive factor. On series-parallel graphs, however, we present a pseudo-polynomial time dynamic programming algorithm for the problem. We then study the case that the minimum quantity is the same for each arc in the network and show that, under this restriction, the problem is still weakly NP-complete on general graphs, but can be solved in strongly polynomial time on series-parallel graphs. On general graphs, we present a \((2 - 1/\lambda) \)-approximation algorithm for this case, where \(\lambda\) denotes the common minimum quantity of all arcs.

- 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.

- 141
- Changepoint tests for INARCH time series (2011)
- In this paper, we discuss the problem of testing for a changepoint in the structure of an integer-valued time series. In particular, we consider a test statistic of cumulative sum (CUSUM) type for general Poisson autoregressions of order 1. We investigate the asymptotic behaviour of conditional least-squares estimates of the parameters in the presence of a changepoint. Then, we derive the asymptotic distribution of the test statistic under the hypothesis of no change, allowing for the calculation of critical values. We prove consistency of the test, i.e. asymptotic power 1, and consistency of the corresponding changepoint estimate. As an application, we have a look at changepoint detection in daily epileptic seizure counts from a clinical study.

- 140
- Variants of the Shortest Path Problem (2011)
- The shortest path problem in which the \((s,t)\)-paths \(P\) of a given digraph \(G =(V,E)\) are compared with respect to the sum of their edge costs is one of the best known problems in combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, \(k\)-sum and \(k\)-max objectives, \((k_1, k_2)-max, (k_1, k_2)\)-balanced and several types of trimmed-mean objectives. We give a survey on existing algorithms and propose a general model for those problems not yet treated in literature. The latter is based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial time if the given graph is acyclic and the number of resources is fixed. In our setting, however, these problems can be solved in strongly polynomial time. Combining this with known results on \(k\)-sum and \(k\)-max optimization for general combinatorial problems, we obtain strongly polynomial algorithms for a variety of path problems on acyclic and general digraphs.

- 139a
- Asymptotic Order of the Parallel Volume Difference (2012)
- In this paper we investigate the asymptotic behaviour of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance \(r\) tends to infinity. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in a \(d\)-dimensional space. Then we will show that in Euclidean spaces this difference can at most have order \(r^{d-3}\). These results have several applications, e.g. we will use them to compute the derivative of \(f_\mu(rK)\) in \(r = 0\), where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a body and \(rK\) is the Minkowski-product of \(r\) and \(K\). Finally we present applications concerning Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.

- 139
- Asymptotic order of the parallel volume difference (2011)
- In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in [13]. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in dimension \(d\). Then we will show that in the Euclidean case this difference can at most have order \(r^{d-3}\). We will also examine the asymptotic behavior of the derivative of this difference as the distance tends to infinity. After this we will compute the derivative of \(f_\mu (rK)\) in \(r\), where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a fixed body and \(rK\) is the Minkowski-product of \(r\) and \(K\). Finally we will use these results to examine Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.

- 138
- A uniform central limit theorem for neural network based autoregressive processes with applications to change-point analysis (2011)
- We consider an autoregressive process with a nonlinear regression function that is modeled by a feedforward neural network. We derive a uniform central limit theorem which is useful in the context of change-point analysis. We propose a test for a change in the autoregression function which - by the uniform central limit theorem - has asymptotic power one for a large class of alternatives including local alternatives.

- 137
- Testing for parameter stability in nonlinear autoregressive models (2011)
- In this paper we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure we model 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, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change-point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log-returns is given.