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Finding "good" cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP hard in general graphs. This result holds both for the variable cardinality case (i.e. all cycles of the graph are considered) and the fixed cardinality case (i.e. only cycles with a given cardinality p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for instance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fixed to be a rectangle one can analyse rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinality, and variable cardinality. In all cases a com plete characterization of the optimal cycles and closed form expressions of the optimal objective values are given, yielding polynomial time algorithms for all cases of center rectangle problems. Finally, it is shown that center cycles can be chosen as rectangles for small cardinalities such that the center cycle problem in grid graphs is in these cases completely solved.

For some decades radiation therapy has been proved successful in cancer treatment. It is the major task of clinical radiation treatment planning to realise on the one hand a high level dose of radiation in the cancer tissue in order to obtain maximum tumour control. On the other hand it is obvious that it is absolutely necessary to keep in the tissue outside the tumour, particularly in organs at risk, the unavoidable radiation as low as possible. No doubt, these two objectives of treatment planning high level dose in the tumour, low radiation outside the tumour have a basically contradictory nature. Therefore, it is no surprise that inverse mathematical models with dose distribution bounds tend to be infeasible in most cases. Thus, there is need for approximations compromising between overdosing the organs at risk and underdosing the target volume. Differing from the currently used time consuming iterative approach, which measures deviation from an ideal (non-achievable) treatment plan using recursively trial-and-error weights for the organs of interest, we go a new way trying to avoid a priori weight choices and consider the treatment planning problem as a multiple objective linear programming problem: with each organ of interest, target tissue as well as organs at risk, we associate an objective function measuring the maximal deviation from the prescribed doses. We build up a data base of relatively few efficient solutions representing and approximating the variety of Pareto solutions of the multiple objective linear programming problem. This data base can be easily scanned by physicians looking for an adequate treatment plan with the aid of an appropriate online tool.

Given a railway network together with information on the population and their use of the railway infrastructure, we are considering the e ffects of introducing new train stops in the existing railway network. One e ffect concerns the accessibility of the railway infrastructure to the population, measured in how far people live from their nearest train stop. The second effect we study is the change in travel time for the railway customers that is induced by new train stops. Based on these two models, we introduce two combinatorial optimization problems and give NP-hardness results for them. We suggest an algorithmic approach for the model based on travel time and give first experimental results.

Finding a delivery plan for cancer radiation treatment using multileaf collimators operating in ''step-and-shoot mode'' can be formulated mathematically as a problem of decomposing an integer matrix into a weighted sum of binary matrices having the consecutive-ones property - and sometimes other properties related to the collimator technology. The efficiency of the delivery plan is measured by both the sum of weights in the decomposition, known as the total beam-on time, and the number of different binary matrices appearing in it, referred to as the cardinality, the latter being closely related to the set-up time of the treatment. In practice, the total beam-on time is usually restricted to its minimum possible value, (which is easy to find), and a decomposition that minimises cardinality (subject to this restriction) is sought.

In this paper we generalize the classical shortest path problem in two ways. We consider two objective functions and time-dependent data. The resulting problem, called the time-dependent bicriteria shortest path problem (TdBiSP), has several interesting practical applications, but has not gained much attention in the literature.

Selection of new projects is one of the major decision making activities in any company. Given a set of potential projects to invest, a subset which matches the company's strategy and internal resources best has to be selected. In this paper, we propose a multicriteria model for portfolio selection of projects, where we take into consideration that each of the potential projects has several - usually conflicting - values.

Multileaf Collimators (MLC) consist of (currently 20-100) pairs of movable metal leaves which are used to block radiation in Intensity Modulated Radiation Therapy (IMRT). The leaves modulate a uniform source of radiation to achieve given intensity profiles. The modulation process is modeled by the decomposition of a given non-negative integer matrix into a non-negative linear combination of matrices with the (strict) consecutive ones property.

Given an undirected, connected network G = (V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as its graph theoretic counterpart, the cycle basis problem. We consider two versions of the problem, the unconstrained and the fundamental cut basis problem. For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem and is thus solvable in strongly polynomial time. The complexity of this algorithm improves the complexity of the best algorithms for the cycle basis problem, such that it is preferable for cycle basis problems in planar graphs. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each, from a spanning tree T is shown to be NP-hard. We present heuristics, integer programming formulations and summarize first experiences with numerical tests.

While in classical scheduling theory the locations of machines are assumed to be fixed we will show how to tackle location and scheduling problems simultaneously. Obviously, this integrated approach enhances the modeling power of scheduling for various real-life problems. In this paper, we present in an exemplary way theory and a solution algorithm for a specific type of a scheduling and a rather general, planar location problem, respectively. More general results and a report on numerical tests will be presented in a subsequent paper.