René Weiskircher

An Algorithmic Framework for the Exact Solution of the Prize-Collecting Steiner Tree Problem

The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way.Our main contribution is the formulation and implementation of a branch-and-cut algorithm based on a directed graph model where we combine several state-of-the-art methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems.We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of large-scale real-world instances arising in the design of fiber optic networks, we also obtain optimal solution values.

Inserting an edge into a planar graph

Abstract Computing a crossing minimum drawing of a given planar graph G augmented by an additional edge e where all crossings involve e, has been a long standing open problem in graph drawing. Alternatively, the problem can be stated as finding a combinatorial embedding of a planar graph G where the given edge e can be inserted with the minimum number of crossings. Many problems concerned with the optimization over the set of all combinatorial embeddings of a planar graph turned out to be NP-hard. Surprisingly, we found a conceptually simple linear time algorithm based on SPQR-trees, that is able to find a solution with the minimum number of crossings.

Combining a Memetic Algorithm with Integer Programming to Solve the Prize-Collecting Steiner Tree Problem

The prize-collecting Steiner tree problem on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. For this well-known problem we develop a new algorithmic framework consisting of three main parts:

Robustness and Resilience

Intuitively, a complex network is robust if it keeps its basic functionality even under failure of some of its components. The study of robustness in networks is important because a thorough understanding of the behavior of certain classes of networks under failures and attacks may help to protect, for instance, communication networks like the Internet against assaults or to exploit weaknesses of metabolic networks in drug design.

The Fractional Prize-Collecting Steiner Tree Problem on Trees

We consider the fractional prize-collecting Steiner tree problem on trees. This problem asks for a subtree T containing the root of a given tree G=(V,E) maximizing the ratio of the vertex profits ∑ v ∈ V (T) p(v) and the edge costs ∑ e ∈ E(T) c(e) plus a fixed cost c 0 and arises in energy supply management. We experimentally compare three algorithms based on parametric search: the binary search method, Newton’s method, and a new algorithm based on Megiddo’s parametric search method. We show improved bounds on the running time for the latter two algorithms. The best theoretical worst case running time, namely O(|V|log|V|), is achieved by our new algorithm. A surprising result of our experiments is the fact that the simple Newton method is the clear winner of the tested algorithms.

Subgraph Induced Planar Connectivity Augmentation

Given a planar graph G =( V, E) and a vertex set W ⊆ V , the subgraph induced planar connectivity augmentation problem asks for a minimum cardinality set F of additional edges with end vertices in W such that G =( V, E∪F ) is planar and the subgraph of G induced by W is connected. The problem arises in automatic graph drawing in the context of c-planarity testing of clustered graphs. We describe a linear time algorithm based on SPQR-trees that tests if a subgraph induced planar connectivity augmentation exists and, if so, constructs a minimum cardinality augmenting edge set.

A multi-agent system for decentralised fractional shared resource constraint scheduling

We consider a collaborative scheduling problem motivated by mining in remote off-grid areas. In our model, jobs are preassigned to processors who have their own machine for executing them. Because each job needs a certain amount of a resource shared between the processors, a coordination mechanism between the processors is needed. We present a framework which collaboratively computes a schedule while exchanging only limited information between the processors and a central resource manager. Our computational experiments show that our negotiated approach outperforms a one-shot solution approach by a wide margin and produces fairer solutions than a centralised genetic algorithm that can make use of the private information of each processor. Depending on the number of processors, the solution quality found by the mechanism presented in this paper is competitive with or even better than that of the centralised genetic algorithm.

Using agents for solving a multi-commodity-flow problem

We investigate a commodity trading problem in a flow network with arbitrary topology where sinks combine commodities into bundles in order to generate profits. Our focus is the profit maximization problem for the trading network under both central and distributed control. We compute solutions for the central control problem using an integer linear program while we compute solutions for the distributed case by implementing the nodes in the network as software-agents that exchange messages in order to establish profitable trades. We report on computational results using both methods and demonstrate that there is a connection between agent profits and a centrality measure developed for the problem. We also demonstrate that with our current agent strategy, there is a trade-off between the agents acting too quickly before enough information is available and waiting too long and thus giving each agent too much information and thus too much power over the outcome.

Collaborative Resource Constraint Scheduling with a Fractional Shared Resource

We consider a collaborative scheduling problem motivated by mining in remote off-grid areas. In our model, jobs are assigned to processors who each have their own machine for executing them. As each job needs a certain amount of a resource shared between the processors, a coordination mechanism between the processors is needed. We present a framework which collaboratively computes a schedule while exchanging only limited information between the processors and a central resource manager. Our computational experiments show that our negotiated approach outperforms a one-shot solution approach by a wide margin and produces fairer solutions than a centralised genetic algorithm that can make use of the private information of each processor. Depending on the number of processors, the solution quality found by the mechanism presented in this paper is competitive with or even better than that of the centralised genetic algorithm.

Drawing planar graphs

When we want to draw a graph to make the information contained in its structure easily accessible, it is highly desirable to have a drawing with as few edge crossings as possible (Purchase et al., 1997; Purchase, 1997). The class of graphs that can be drawn with no crossings at all is the class of planar graphs. Algorithms for drawing planar graphs are the main subject of this chapter.