Hartmut Noltemeier

Vorono trees and clustering problems

This paper presents a new data structure called Voronoi tree to support the solution of proximity problems in general pseudo metric spaces with efficiently computable distance functions. We analyse some structural properties and report experimental results showing that Voronoi trees are a proper and very efficient tool for the representation of proximity properties and generation of suitable clusterings.

Monotonous Bisector* Trees - A Tool for Efficient Partitioning of Complex Scenes of Geometric Objects

We are concerned with the problem of partitioning complex scenes of geometric objects in order to support the solutions of proximity problems in general metric spaces with an efficiently computable distance function. We present a data structure called Monotonous Bisector Tree, which can be regarded as a divisive hierarchical approach of centralized clustering methods (compare [3] and [12]). We analyze some structural properties showing that Monotonous Bisector Trees are a proper tool for a general representation of proximity information in complex scenes of geometric objects.

Approximation Algorithms for Certain Network Improvement Problems

We study budget constrained network upgrading problems. Such problems aim at finding optimal strategies for improving a network under some cost measure subject to certain budget constraints. Given an edge weighted graph G = (V, E), in the edge based upgrading model, it is assumed that each edge e of the given network also has an associated function ce (t) that specifies the cost of upgrading the edge by an amount t. A reduction strategy specifies for each edge e the amount by which the length ℓ(e) is to be reduced. In the node based upgrading model, a node v can be upgraded at an expense of c(v). Such an upgrade reduces the delay of each edge incident on v. For a given budget B, the goal is to find an improvement strategy such that the total cost of reduction is at most the given budget B and the cost of a subgraph (e.g. minimum spanning tree) under the modified edge lengths is the best over all possible strategies which obey the budget constraint.After providing a brief overview of the models and definitions of the various problems considered, we present several new results on the complexity and approximability of network improvement problems.

Modifying edges of a network to obtain short subgraphs

Abstract This paper considers problems of the following type: We are given an edge weighted graph G = ( V , E ). It is assumed that each edge e of the given network has an associated function c e that specifies the cost of shortening the edge by a given amount and that there is a budget B on the total reduction cost. The goal is to develop a reduction strategy satisfying the budget constraint so that the total length of a minimum spanning tree in the modified network is the smallest possible over all reduction strategies that obey the budget constraint. We show that in general the problem of computing an optimal reduction strategy for modifying the network as above is NP-hard even for simple classes of graphs and linear functions c e . We present the first polynomial time approximation algorithms for the problem, where the cost functions c e are allowed to be taken from a broad class of functions. We also present improved approximation algorithms for the class of treewidth-bounded graphs when the cost functions are linear. Our results can be extended to obtain approximation algorithms for more general network design problems such as Steiner trees and generalized Steiner networks.

Multiple voting location and single voting location on trees

Abstract We examine voting location problems in which the goal is to place, based on an election amongst the users, a given number of facilities in a graph. The user preference is modeled by shortest path distances in the graph. A Condorcet solution is a set of facilities to which there does not exist an alternative set preferred by a majority of the users. Recent works generalize the model to additive indifference and replaced user majority by γ -proportion. We show that for multiple voting location, Condorcet and Simpson decision problems are Σ 2 p -complete, and investigate the approximability of the Simpson and the Simpson score optimization problem. Further we contribute a result towards lower bounds on the complexity of the single voting location problem. On the positive side we develop algorithms for the optimization problems on tree networks which are substantially faster than the existing algorithms for general graphs. Finally we suggest a generalization of the indifference notion to threshold functions.

Compact location problems

We consider the problem of placing a specified number (p) of facilities on the nodes of a network so as to minimize some measure of the distances between facilities. This formulation models a number of problems arising in facility location, statistical clustering, pattern recognition, and also a processor allocation problem in multiprocessor systems.

Improving Minimum Cost Spanning Trees by Upgrading Nodes

We study budget constrained network upgrading problems. We are given an undirected edge-weighted graph G=(V,E), where node v?V can be upgraded at a cost of c(v). This upgrade reduces the weight of each edge incident on v. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a minimum spanning tree of weight no more than a given budget D. The results obtained in the paper include?On the positive side, we provide a polynomial time approximation algorithm for the above upgrading problem when the difference between the maximum and minimum edge weights is bounded by a polynomial in , the number of nodes in the graph. The solution produced by the algorithm satisfies the budget constraint, and the cost of the upgrading set produced by the algorithm is O(log) times the minimum upgrading cost needed to obtain a spanning tree of weight at most .?In contrast, we show that, unless ?(), there can be no polynomial time approximation algorithm for the problem that produces a solution with upgrading cost at most ?

Improving spanning trees by upgrading nodes

We study budget constrained optimal network upgrading problems. We are given an edge weighted graph G=(V, E) where node υ ∈ V can be upgraded at a cost of c(υ). This upgrade reduces the delay of each link emanating from υ. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a good performance. We consider two performance measures, namely, the weight of a minimum spanning tree and the bottleneck weight of a minimum bottleneck spanning tree, and present approximation algorithms.

Network design and improvement

Inspired by the fact that many combinatorial optimization problems arising in practice are NP-hard, the design of efficient approximation algorithms has been a major research topic for the last years. Since we can not expect to solve any NP-hard problem in polynomial time, it is meaningful to compromise optimality of a solution and settle for a “sufficiently good” solution that can be computed efficiently in polynomial time.

A data structure for representing and efficient querying large scenes of geometric objects: MB* trees

We are concerned with the problem of partitioning complex scenes of geometric objects in order to support the solutions of proximity problems in general metric spaces with an efficiently computable distance function. We present a data structure called Monotone Bisector* Tree (MB* Tree), which can be regarded as a divisive hierarchical approach of centralized clustering methods (compare [3] and [10]). We analyze some structural properties showing that MB* Trees are a proper tool for a general representation of proximity information in complex scenes of geometric objects.