We consider various properties of a general parity domination problem: given a graph G, one is lo... more We consider various properties of a general parity domination problem: given a graph G, one is looking for a subset S of the vertex set such that the open/closed neighborhood of each vertex contains an even/odd number of vertices in S (it is prescribed individually for each vertex which of these applies). We define the parameter s(G) to be the number of solvable instances out of 4 n possibilities and study the properties of this parameter. Upper and lower bounds for general graphs and trees are given as well as a remarkable recurrence formula for rooted trees and-more generally-an algorithm for graphs of bounded tree-width. Furthermore, we give explicit formulas in several special cases and investigate random graphs.
Summary This paper is dedicated to location problems on graphs. We propose a linear time algorit... more Summary This paper is dedicated to location problems on graphs. We propose a linear time algorithm for the 1-median problem on wheel graphs. Moreover, some general results for the 1-median problem are summarized and parametric median problems are investigated. These results lead to a solution method for the 2-median problem on cactus graphs, i.e., on graphs where no two cycles have
In this paper, we consider the 1-median problem in Rd with the Chebyshev-norm. We give an optimal... more In this paper, we consider the 1-median problem in Rd with the Chebyshev-norm. We give an optimality criterion for this problem which enables us to solve the following inverse location problem in polynomial time: Given n points P1,…,Pn with non-negative weights wi and a point P0 the task is to find new non-negative weights w˜i such that P0 is a
ABSTRACT In this paper, we consider the 1-median problem in RdRd with the Chebyshev-norm. We give... more ABSTRACT In this paper, we consider the 1-median problem in RdRd with the Chebyshev-norm. We give an optimality criterion for this problem which enables us to solve the following inverse location problem by a combinatorial algorithm in polynomial time: Given nn points P1,…,Pn∈RdP1,…,Pn∈Rd with non-negative weights wiwi and a point P0P0 the task is to find new non-negative weights w̃i such that P0P0 is a 1-median with respect to the new weights and ‖w−w̃‖1 is minimized. In fact, this problem reduces to a 2-balanced flow problem for which an optimal solution can be obtained by solving a fractional bb-matching problem.
Summary This paper is dedicated to location problems on graphs. We propose a linear time algorit... more Summary This paper is dedicated to location problems on graphs. We propose a linear time algorithm for the 1-median problem on wheel graphs. Moreover, some general results for the 1-median problem are summarized and parametric median problems are investigated. These results lead to a solution method for the 2-median problem on cactus graphs, i.e., on graphs where no two cycles have
In this paper we investigate two different models for minimizing the makespan of batch processes ... more In this paper we investigate two different models for minimizing the makespan of batch processes by mixed-integer linear programming models. Special emphasis is laid on a small number of binary variables and on valid constraints. After a reformulation of the objective function, it is for the first time possible to find optimal solutions for medium-sized benchmark problems. Furthermore, a powerful
We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network... more We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series-parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NPhard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series-parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.
We consider the 1-median problem in R d with the Chebyshev norm: given n points with non-negative... more We consider the 1-median problem in R d with the Chebyshev norm: given n points with non-negative weights, find a point that minimizes the sum of the weighted distances to the given points. We propose a combinatorial algorithm for this problem by reformulating it as a fractional b-matching problem. This graph-theoretical problem can be solved by a min-cost-flow algorithm. Moreover, we show that there is a 1-median, which is half-integral, provided that the points have integral coordinates.
Journal of Mathematical Modelling and Algorithms, 2011
Consider the set S of points in the plane consisting of the ordered pairs (i, j), where 1 i m and... more Consider the set S of points in the plane consisting of the ordered pairs (i, j), where 1 i m and 1 j n. A problem related to the study of segmentation evaluation of visual images concerns finding a permutation σ of the points of S for which the sum
This paper deals with facility location problems on graphs with positive and negative vertex weig... more This paper deals with facility location problems on graphs with positive and negative vertex weights. We consider two different objective functions: In the first one (MWD) vertices with positive weight are assigned to the closest facility, whereas vertices with negative weight are assigned to the farthest facility. In the second one (WMD) all the vertices are assigned to the nearest
In this paper we investigate a heuristic for batch processing problems occurring in the chemical ... more In this paper we investigate a heuristic for batch processing problems occurring in the chemical industry, where the objective is to minimize the makespan. Usually, this kind of problem is solved using mixed-integer linear programs. However, due to the large number of binary variables good results within a reasonable computational time could only be obtained for small instances.
In this paper, we consider the Online Target Date Assignment Problem (OnlineTDAP) with deferral t... more In this paper, we consider the Online Target Date Assignment Problem (OnlineTDAP) with deferral time one and unsplittable requests for general downstream problems, where the downstream cost are nonnegative, additive and satisfy the triangle inequality.
We consider various properties of a general parity domination problem: given a graph G, one is lo... more We consider various properties of a general parity domination problem: given a graph G, one is looking for a subset S of the vertex set such that the open/closed neighborhood of each vertex contains an even/odd number of vertices in S (it is prescribed individually for each vertex which of these applies). We define the parameter s(G) to be the number of solvable instances out of 4 n possibilities and study the properties of this parameter. Upper and lower bounds for general graphs and trees are given as well as a remarkable recurrence formula for rooted trees and-more generally-an algorithm for graphs of bounded tree-width. Furthermore, we give explicit formulas in several special cases and investigate random graphs.
Summary This paper is dedicated to location problems on graphs. We propose a linear time algorit... more Summary This paper is dedicated to location problems on graphs. We propose a linear time algorithm for the 1-median problem on wheel graphs. Moreover, some general results for the 1-median problem are summarized and parametric median problems are investigated. These results lead to a solution method for the 2-median problem on cactus graphs, i.e., on graphs where no two cycles have
In this paper, we consider the 1-median problem in Rd with the Chebyshev-norm. We give an optimal... more In this paper, we consider the 1-median problem in Rd with the Chebyshev-norm. We give an optimality criterion for this problem which enables us to solve the following inverse location problem in polynomial time: Given n points P1,…,Pn with non-negative weights wi and a point P0 the task is to find new non-negative weights w˜i such that P0 is a
ABSTRACT In this paper, we consider the 1-median problem in RdRd with the Chebyshev-norm. We give... more ABSTRACT In this paper, we consider the 1-median problem in RdRd with the Chebyshev-norm. We give an optimality criterion for this problem which enables us to solve the following inverse location problem by a combinatorial algorithm in polynomial time: Given nn points P1,…,Pn∈RdP1,…,Pn∈Rd with non-negative weights wiwi and a point P0P0 the task is to find new non-negative weights w̃i such that P0P0 is a 1-median with respect to the new weights and ‖w−w̃‖1 is minimized. In fact, this problem reduces to a 2-balanced flow problem for which an optimal solution can be obtained by solving a fractional bb-matching problem.
Summary This paper is dedicated to location problems on graphs. We propose a linear time algorit... more Summary This paper is dedicated to location problems on graphs. We propose a linear time algorithm for the 1-median problem on wheel graphs. Moreover, some general results for the 1-median problem are summarized and parametric median problems are investigated. These results lead to a solution method for the 2-median problem on cactus graphs, i.e., on graphs where no two cycles have
In this paper we investigate two different models for minimizing the makespan of batch processes ... more In this paper we investigate two different models for minimizing the makespan of batch processes by mixed-integer linear programming models. Special emphasis is laid on a small number of binary variables and on valid constraints. After a reformulation of the objective function, it is for the first time possible to find optimal solutions for medium-sized benchmark problems. Furthermore, a powerful
We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network... more We study the complexity of finding extreme pure Nash equilibria in symmetric (unweighted) network congestion games. In our context best and worst equilibria are those with minimum respectively maximum makespan. On series-parallel graphs a worst Nash equilibrium can be found by a Greedy approach while finding a best equilibrium is NPhard. For a fixed number of users we give a pseudo-polynomial algorithm to find the best equilibrium in series-parallel networks. For general network topologies also finding a worst equilibrium is NP-hard.
We consider the 1-median problem in R d with the Chebyshev norm: given n points with non-negative... more We consider the 1-median problem in R d with the Chebyshev norm: given n points with non-negative weights, find a point that minimizes the sum of the weighted distances to the given points. We propose a combinatorial algorithm for this problem by reformulating it as a fractional b-matching problem. This graph-theoretical problem can be solved by a min-cost-flow algorithm. Moreover, we show that there is a 1-median, which is half-integral, provided that the points have integral coordinates.
Journal of Mathematical Modelling and Algorithms, 2011
Consider the set S of points in the plane consisting of the ordered pairs (i, j), where 1 i m and... more Consider the set S of points in the plane consisting of the ordered pairs (i, j), where 1 i m and 1 j n. A problem related to the study of segmentation evaluation of visual images concerns finding a permutation σ of the points of S for which the sum
This paper deals with facility location problems on graphs with positive and negative vertex weig... more This paper deals with facility location problems on graphs with positive and negative vertex weights. We consider two different objective functions: In the first one (MWD) vertices with positive weight are assigned to the closest facility, whereas vertices with negative weight are assigned to the farthest facility. In the second one (WMD) all the vertices are assigned to the nearest
In this paper we investigate a heuristic for batch processing problems occurring in the chemical ... more In this paper we investigate a heuristic for batch processing problems occurring in the chemical industry, where the objective is to minimize the makespan. Usually, this kind of problem is solved using mixed-integer linear programs. However, due to the large number of binary variables good results within a reasonable computational time could only be obtained for small instances.
In this paper, we consider the Online Target Date Assignment Problem (OnlineTDAP) with deferral t... more In this paper, we consider the Online Target Date Assignment Problem (OnlineTDAP) with deferral time one and unsplittable requests for general downstream problems, where the downstream cost are nonnegative, additive and satisfy the triangle inequality.
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Papers by Johannes Hatzl