Hans-Christoph Wirth

On the minimum label spanning tree problem

We study the Minimum Label Spanning Tree Problem. In this problem, we are given an undirected graph whose edges are labeled with colors. The goal is to find a spanning tree which uses as little different colors as possible. We present an approximation algorithm with logarithmic performance guarantee. On the other hand, our hardness results show that the problem cannot be approximated within a constant factor.

(r,p)-centroid problems on paths and trees

An instance of the (r,p)-centroid problem is given by an edge and node weighted graph. Two competitors, the leader and the follower, are allowed to place p and r facilities, respectively, into the graph. Users at the nodes connect to the closest facility. A solution of the (r,p)-centroid problem is a leader placement such that the maximum total weight of the users connecting to any follower placement is as small as possible. We show that the absolute (r,p)-centroid problem is NP-hard even on a path which answers a long-standing open question of the complexity of the problem on trees (Hakimi, 1990 [10]). Moreover, we provide polynomial time algorithms for the discrete (r,p)-centroid on paths and the (1,p)-centroid on trees, and complementary hardness results for more complex graph classes.

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.

Reload cost problems: minimum diameter spanning tree

We examine a network design problem under the reload cost model. Given an undirected edge colored graph, reload costs on a path arise at a node where the path uses consecutive edges of different colors. We consider the problem of finding a spanning tree of minimum diameter with respect to the reload costs. We present lower bounds for the approximability even on graphs with maximum degree 5. On the other hand we provide an exact algorithm for graphs of maximum degree 3.

Multiple Hotlink Assignment

The input for the hotlink assignment problem consists of a node weighted directed acyclic graph with a designated root node r. The goal is to minimize the weighted shortest path length rooted at r by adding a restricted number of outgoing arcs (hotlinks) to each node. The (h, k)-hotlink assignment problem is defined on k-regular complete trees, and at most h hotlinks can be assigned to each node. We contribute algorithms for the (1, k), (2, k), and (k-1, k) hotlink assignment problem.

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 ?

Euler is standing in line dial-a-ride problems with precedence-constraints

Abstract In this paper we study algorithms for “Dial-a-Ride” transportation problems. In the basic version of the problem we are given transportation jobs between the vertices of a graph and the goal is to find a shortest transportation that serves all the jobs. This problem is known to be NP -hard even on trees. We consider the extension when precedence relations between the jobs with the same source are given. Our results include a polynomial time algorithm on paths and approximation algorithms for general graphs and trees with performances of 9/4 and 5/3, respectively.

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.

Budgeted Maximum Graph Coverage

An instance of the maximum coverage problem is given by a set of weighted ground elements and a cost weighted family of subsets of the ground element set. The goal is to select a subfamily of total cost of at most that of a given budget maximizing the weight of the covered elements.We formulate the problem on graphs: In this situation the set of ground elements is specified by the nodes of a graph, while the family of covering sets is restricted to connected subgraphs. We show that on general graphs the problem is polynomial time solvable if restricted to sets of size at most 2, but becomes NP-hard if sets of size 3 are permitted. On trees, we prove polynomial time solvability if each node appears in a fixed number of sets. In contrast, if vertices are allowed to appear an unbounded number of times, the problem is NP-hard even on stars. We finally give a polynomial time algorithm for the special case where a star is covered by paths.

Upgrading Bottleneck Constrained Forests

We study bottleneck constrained network upgrading problems. We are given an edge weighted graph G=(V,E) where node v ∈ V can be upgraded at a cost of c(v). This upgrade reduces the delay of each link emanating from v. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a good performance. The performance is measured by the bottleneck weight of a constrained forest defined by a proper function [GW95]. These problems are a generalization of the node weighted constrained forest problems studied by Klein and Ravi [KR95].