Ekkehard Köhler

Optimal FPGA module placement with temporal precedence constraints

We consider the optimal placement of hardware modules in space and time for FPGA architectures with reconfiguration capabilities, where modules are modeled as three-dimensional boxes in space and time. Using a graph-theoretic characterization of feasible packings, we are able to solve the following problems. (a) Find the minimal execution time of the given problem on an FPGA of fixed size, (b) Find the FPGA of minimal size to accomplish the tasks within a fired time limit. Furthermore, our approach is perfectly suited for the treatment of precedence constraints for the sequence of tasks, which are present in virtually all practical instances. Additional mathematical structures are developed that lead to a powerful framework for completing optimal solutions. The usefulness is illustrated by computational results.

Acceleration of shortest path and constrained shortest path computation

We study acceleration methods for point-to-point shortest path and constrained shortest path computations in directed graphs, in particular in road and railroad networks. Our acceleration methods are allowed to use a preprocessing of the network data to create auxiliary information which is then used to speed-up shortest path queries. We focus on two methods based on Dijkstras algorithm for shortest path computations and two methods based on a generalized version of Dijkstra for constrained shortest paths. The methods are compared with other acceleration techniques, most of them published only recently. We also look at appropriate combinations of different methods to find further improvements. For shortest path computations we investigate hierarchical multiway-separator and arc-flag approaches. The hierarchical multiway-separator approach divides the graph into regions along a multiway-separator and gathers information to improve the search for shortest paths that stretch over several regions. A new multiway-separator heuristic is presented which improves the hierarchical separator approach. The arc-flag approach divides the graph into regions and gathers information on whether an arc is on a shortest path into a given region. Both methods yield significant speed-ups of the plain Dijkstras algorithm. The arc flag method combined with an appropriate partition and a bi-directed search achieves an average speed-up of up to 1,400 on large networks. This combination narrows down the search space of Dijkstras algorithm to almost the size of the corresponding shortest path for long distance shortest path queries. For the constrained shortest path problem we show that goal-directed and bi-directed acceleration methods can be used both individually and in combination. The goal-directed search achieves the best speed-up factor of 110 for the constrained problem.

On the k-Splittable Flow Problem

In traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, no restriction is imposed on the number of paths used for delivering each commodity; it is thus feasible to spread the flow over a large number of different paths. Motivated by routing problems arising in real-life applications, such as, e. g., telecommunication, unsplittable flows have moved into the focus of research. Here, the demand of each commodity may not be split but has to be sent along a single path.In this paper, a generalization of this problem is studied. In the considered flow model, a commodity can be split into a bounded number of chunks which can then be routed on different paths. In contrast to classical (splittable) flows and unsplittable flows, already the single-commodity case of this problem is NP-hard and even hard to approximate. We present approximation algorithms for the single- and multicommodity case and point out strong connections to unsplittable flows. Moreover, results on the hardness of approximation are presented. It particular, we show that some of our approximation results are in fact best possible, unless P=NP.

The k-Splittable Flow Problem

AbstractIn traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However, no restriction is imposed on the number of paths used for delivering each commodity; it is thus feasible to spread the flow over a large number of different paths. Motivated by routing problems arising in real-life applications, e.g., telecommunication, unsplittable flows have moved into the focus of research. Here, the demand of each commodity may not be split but has to be sent along a single path. In this paper a generalization of this problem is studied. In the considered flow model, a commodity can be split into a bounded number of chunks which can then be routed on different paths. In contrast to classical (splittable) flows and unsplittable flows, the single-commodity case of this problem is already NP-hard and even hard to approximate. We present approximation algorithms for the single- and multi-commodity case and point out strong connections to unsplittable flows. Moreover, results on the hardness of approximation are presented. In particular, we show that some of our approximation results are in fact best possible, unless P = NP.

Dynamic Routing of Automated Guided Vehicles in Real-time

Automated Guided Vehicles (AGVs) are state-of-the-art technology for optimizing large scale production systems and are used in a wide range of application areas. A standard task in this context is to find efficient routing schemes, i.e., algorithms that route these vehicles through the particular environment. The productivity of the AGVs is highly dependent on the used routing scheme. In this work we study a particular routing algorithm for AGVs in an automated logistic system. For the evaluation of our algorithm we focus on Container Terminal Altenwerder (CTA) at Hamburg Harbor. However, our model is appropriate for an arbitrary graph. The key feature of this algorithm is that it avoids collisions, deadlocks and livelocks already at the time of route computation (conflict-free routing), whereas standard approaches deal with these problems only at the execution time of the routes. In addition, the algorithm considers physical properties of the AGVs and certain safety aspects implied by the particular application.

An Approximation Algorithm for the Tree t-Spanner Problem on Unweighted Graphs via Generalized Chordal Graphs

A spanning tree T of a graph G is called a treet-spanner of G if the distance between every pair of vertices in T is at most t times their distance in G. In this paper, we present an algorithm which constructs for an n-vertex m-edge unweighted graph G: a tree (2⌊log2n⌋)-spanner in O(mlogn) time, if G is a chordal graph (i.e., every induced cycle of G has length 3);a tree (2ρ⌊log2n⌋)-spanner in O(mnlog2n) time or a tree (12ρ⌊log2n⌋)-spanner in O(mlogn) time, if G is a graph admitting a Robertson-Seymour’s tree-decomposition with bags of radius at most ρ in G; anda tree (2⌈t/2⌉⌊log2n⌋)-spanner in O(mnlog2n) time or a tree (6t⌊log2n⌋)-spanner in O(mlogn) time, if G is an arbitrary graph admitting a tree t-spanner. For the latter result we use a new necessary condition for a graph to have a tree t-spanner: if a graph G has a tree t-spanner, then G admits a Robertson-Seymour’s tree-decomposition with bags of radius at most ⌈t/2⌉ and diameter at most t in G.

Linear Time Algorithms for Hamiltonian Problems on (Claw,Net)-Free Graphs

We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that 2-connected claw-free graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the input is restricted to (claw,net)-free graphs. These graphs enjoy those interesting structural properties.

On the power of BFS to determine a graph's diameter

Recently, considerable effort has been spent on showing that Lexicographic Breadth First Search (LBFS) can be used to determine a tight bound on the diameter of graphs from various restricted classes. In this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximum-sized cycle that may appear as an induced subgraph. We show that, on graphs that have no induced cycle of size greater than k, BFS finds an estimate of the diameter that is no worse than diam(G) − ⌊k/2⌋.

Length-bounded cuts and flows

An L-length-bounded cut in a graph G with source s, and sink t is a cut that destroys all s-t-paths of length at most L. An L-length-bounded flow is a flow in which only flow paths of length at most L are used. We show that the minimum length-bounded cut problem in graphs with unit edge lengths is

Higher-Dimensional Packing with Order Constraints

\mathcal{NP}