Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Frauke Liers, Christoph Buchheim, Laura Sanità
Universität zu Köln, Technische Universität Dortmund, École Polytechnique Fédérale de Lausanne
An exact Algorithm for Robust Network Design with a Discrete Set of Traffic Matrices
Modern life heavily relies on communication networks that operate efficiently. A crucial issue for the design
of communication networks is robustness with respect to traffic fluctuations, since they often lead to
congestion and traffic bottlenecks. In this paper, we address an NP-hard single commodity robust network
design problem, where the traffic demands change over time. For k different times of the day, we are given
for each node the amount of single-commodity flow it wants to send or to receive. The task is to determine
the minimum-cost edge capacities such that the flow can be routed integrally through the net at all times.
We present an exact branch-and-cut algorithm, based on a decomposition into biconnected network components,
a clever primal heuristic for generating feasible solutions from the linear-programming relaxation,
and a general cutting-plane separation routine that is based on projection and lifting. By presenting extensive
experimental results on realistic instances from the literature, we show that a suitable combination of
these algorithmic components can solve most of these instances to optimality. Furthermore, cutting-plane
separation considerably improves the algorithmic performance.
Literatur
Buchheim, Liers, Sanità (2011). In J. Pahl, T. Reiners and S. Voß (Eds.): INOC 2011, LNCS 6701, pp.
7-17, 2011.
Gregor Pardella, Frauke Liers
Universität zu Köln
Maximum Flows in Grid Graphs
Maximum-flow problems occur in a wide range of applications. Although already well-studied, they are
still an area of active research. The fastest available implementations for determining maximum flows in
graphs are either based on augmenting-path or on push-relabel algorithms. In this talk, we present two
ingredients that, appropriately used, can considerably speed up these methods. On the one hand, we
present flow-conserving conditions under which subgraphs can be contracted to a single vertex. These
rules are in the same spirit as presented by Padberg and Rinaldi (Math. Programming (47), 1990) for
the minimum cut problem in graphs. On the other hand, we propose a two-step max-flow algorithm for
solving the problem on instances coming from physics and computer vision. In the two-step algorithm
flow is first sent along augmenting paths of restricted lengths only. Starting from this flow, the problem is
then solved to optimality using some known max-flow methods. By extensive experiments on instances
coming from applications in theoretical physics and in computer vision and on random instances, we
show that a suitable combination of the proposed techniques speeds up traditionally used methods.
Literatur
Y. Boykov and V. Kolmogorov (2004). An Experimental Comparison of Min-Cut/Max-Flow Algorithms for
Energy Minimization in Vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(9),
1124 – 1137.
A. V. Goldberg and R. E. Tarjan (1988). A New Approach to the Maximum-Flow Problem. Journal of the
Association for Computing Machinery, 35(4), 921 – 940.
M. Padberg and G. Rinaldi (1990). An Efficient Algorithm for the Minimum Capacity Cut Problem. Mathematical
Programming A, 47(1), 19 – 36.
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