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, 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.

Motivated by applications in road traffic control, we study flows in networks featuring special characteristics. Firstly, there are transit times on the arcs of the network which specify the amount of time it takes for flow to travel through an arc; in particular, flow values on arcs may change over time. Secondly, the transit time of an arc varies with the current amount of flow using this arc. The latter feature is crucial for various real-life applications; yet, it dramatically increases the degree of difficulty of the resulting optimization problems. While almost all flow problems with constant transit times on the arcs can be solved efficiently by applying classical (static) flow algorithms in a corresponding time-expanded network, no such approach was known for flow-dependent transit times, up to now. One main contribution of this paper is a time-expanded network with flowdependent transit times to which the whole algorithmic toolbox developed for static flows can be applied. Although this approach does not entirely capture the behavior of flows over time with flowdependent transit times, we present approximation results which provide evidence of its surprising quality.

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For a given number L, an L-length-bounded edge-cut (node-cut, resp.) in a graph G with source s and sink t is a set C of edges (nodes, resp.) such that no s-t-path of length at most L remains in the graph after removing the edges (nodes, resp.) in C. An L-length-bounded flow is a flow that can be decomposed into flow paths of length at most L. In contrast to the classical flow theory, we describe instances for which the minimum L-length-bounded edge-cut (node-cut, resp.) is Θ(n 2/3 )-times (Θ( √ n)-times, resp.) larger than the maximum L-length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is N P-hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of nodecuts and for L ≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O(min{L, n/L}) ⊆ O( √ n) in the node case and O(min{L, n 2 /L 2 , √ m}) ⊆ O(n 2/3 ) in the edge case, where m denotes the number of edges. Concerning L-length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is N P-complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.

Abstract. We present a number of improvements of the basic variant of the arc-flag acceleration (Lauther, 1997(Lauther, , 2004 for point-to-point (P2P) shortest path computations on large graphs. Arc-flags are a modification to the standard Dijkstra algorithm and are used to avoid exploring unnecessary paths during shortest path computation. We assume that for the same input graph the shortest path problem has to be solved repeatedly for different node pairs. Thus, precomputing the arc-flags is possible. We show that the improved arc-flag acceleration achieves speedups of P2P shortest path queries of more than 1,470 on a subnetwork of the German road network 1 with 1M node and 2.5M arcs using 450 bits of additional information per arc. The acceleration factors increase with the size of the input graph. Finally, we present an improved preprocessing version which allows precomputing arc-flags for European and North-American road networks within hours.

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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 Dijkstra's 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 Dijkstra's 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 Dijkstra's 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.

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