Martin Groß

Generalized maximum flows over time

Flows over time and generalized flows are two advanced network flow models of utmost importance, as they incorporate two crucial features occurring in numerous real-life networks. Flows over time feature time as a problem dimension and allow to realistically model the fact that commodities (goods, information, etc.) are routed through a network over time. Generalized flows allow for gain/loss factors on the arcs that model physical transformations of a commodity due to leakage, evaporation, breeding, theft, or interest rates. Although the latter effects are usually time-bound, generalized flow models featuring a temporal dimension have never been studied in the literature. In this paper we introduce the problem of computing a generalized maximum flow over time in networks with both gain factors and transit times on the arcs. While generalized maximum flows and maximum flows over time can be computed efficiently, our combined problem turns out to be NP-hard and even completely non-approximable. A natural special case is given by lossy networks where the loss rate per time unit is identical on all arcs. For this case we present a (practically efficient) FPTAS.

Many-to-one Matchings with Lower Quotas: Algorithms and Complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph \(G= (A \dot{\cup }P, E)\) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (wmlq), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project.

Graph Orientation and Flows over Time

Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems – streets, tracks, etc. – are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem.

A Local-Search Algorithm for Steiner Forest

In the Steiner Forest problem, we are given a graph and a collection of source-sink pairs, and the goal is to find a subgraph of minimum total length such that all pairs are connected. The problem is APX-Hard and can be 2-approximated by, e.g., the elegant primal-dual algorithm of Agrawal, Klein, and Ravi from 1995. We give a local-search-based constant-factor approximation for the problem. Local search brings in new techniques to an area that has for long not seen any improvements and might be a step towards a combinatorial algorithm for the more general survivable network design problem. Moreover, local search was an essential tool to tackle the dynamic MST/Steiner Tree problem, whereas dynamic Steiner Forest is still wide open. It is easy to see that any constant factor local search algorithm requires steps that add/drop many edges together. We propose natural local moves which, at each step, either (a) add a shortest path in the current graph and then drop a bunch of inessential edges, or (b) add a set of edges to the current solution. This second type of moves is motivated by the potential function we use to measure progress, combining the cost of the solution with a penalty for each connected component. Our carefully-chosen local moves and potential function work in tandem to eliminate bad local minima that arise when using more traditional local moves.

Scheduling maintenance jobs in networks

Abstract We investigate the problem of scheduling the maintenance of edges in a network, motivated by the goal of minimizing outages in transportation or telecommunication networks. We focus on maintaining connectivity between two nodes over time; for the special case of path networks, this is related to the problem of minimizing the busy time of machines. We show that the problem can be solved in polynomial time in arbitrary networks if preemption is allowed. If preemption is restricted to integral time points, the problem is NP-hard and in the non-preemptive case we give strong non-approximability results. Furthermore, we give tight bounds on the power of preemption, that is, the maximum ratio of the values of non-preemptive and preemptive optimal solutions. Interestingly, the preemptive and the non-preemptive problem can be solved efficiently on paths, whereas we show that mixing both leads to a weakly NP-hard problem that allows for a simple 2-approximation.

Scheduling Maintenance Jobs in Networks

We investigate the problem of scheduling the maintenance of edges in a network, motivated by the goal of minimizing outages in transportation or telecommunication networks. We focus on maintaining connectivity between two nodes over time; for the special case of path networks, this is related to the problem of minimizing the busy time of machines.

General Bounds for Incremental Maximization

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value

Kreuzzahlrätsel: Sudokus waren gestern

k\in\mathbb{N}

On the use of network flow techniques for assigning evacuees to exits

that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all

On the complexity of instationary gas flows

k