Showing papers by "Rolf H. Möhring published in 2007"

Partitioning graphs to speedup Dijkstra's algorithm

Abstract: We study an acceleration method for point-to-point shortest-path computations in large and sparse directed graphs with given nonnegative arc weights. The acceleration method is called the arc-flag approach and is based on Dijkstra's algorithm. In the arc-flag approach, we allow a preprocessing of the network data to generate additional information, which is then used to speedup shortest-path queries. In the preprocessing phase, the graph is divided into regions and information is gathered on whether an arc is on a shortest path into a given region. The arc-flag method combined with an appropriate partitioning and a bidirected search achieves an average speedup factor of more than 500 compared to the standard algorithm of Dijkstra on large networks (1 million nodes, 2.5 million arcs). This combination narrows down the search space of Dijkstra's algorithm to almost the size of the corresponding shortest path for long-distance shortest-path queries. We conduct an experimental study that evaluates which partitionings are best suited for the arc-flag method. In particular, we examine partitioning algorithms from computational geometry and a multiway arc separator partitioning. The evaluation was done on German road networks. The impact of different partitions on the speedup of the shortest path algorithm are compared. Furthermore, we present an extension of the speedup technique to multiple levels of partitions. With this multilevel variant, the same speedup factors can be achieved with smaller space requirements. It can, therefore, be seen as a compression of the precomputed data that preserves the correctness of the computed shortest paths.

Solutions to real-world instances of PSPACE-complete stacking

Abstract: We investigate a complex stacking problem that stems from storage planning of steel slabs in integrated steel production. Besides the practical importance of such stacking tasks, they are appealing from a theoretical point of view. We show that already a simple version of our stacking problem is PSPACE-complete. Thus, fast algorithms for computing provably good solutions as they are required for practical purposes raise various algorithmic challenges. We describe an algorithm that computes solutions within 5/4 of optimality for all our real-world test instances. The basic idea is a search in an exponential state space that is guided by a state-valuation function. The algorithm is extremely fast and solves practical instances within a few seconds. We assess the quality of our solutions by computing instance-dependent lower bounds from a combinatorial relaxation formulated as mixed integer program. To the best of our knowledge, this is the first approach that provides quality guarantees for such problems.