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For a given number L , an L -length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s - t -path of length at most L remains in the graph after removing the edges (nodes, respectively) 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 classical flow theory, we describe instances for which the minimum L -length-bounded edge-cut (node-cut, respectively) is Θ( n 2/3 )-times (Θ(√ n )-times, respectively) 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 NP -hard to approximate within a factor of 1.1377 for L ≥ 5 in the case of node-cuts 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 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 NP -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.

We study a planning problem arising in SDH/WDM multilayer telecommunication network design. The goal is to find a minimum cost installation of link and node hardware of both network layers such that traffic demands can be realized via grooming and a survivable routing. We present a mixed-integer programming formulation for a predefined set of admissible logical links that takes many practical side constraints into account, including node hardware, several bit rates, and survivability against single physical node or link failures. This model is solved using a branch-and-cut approach with problem-specific preprocessing, MIPbased heuristics, and cutting planes based on either of the two layers. On several realistic two-layer planning scenarios, we show that these ingredients can be very useful to reduce the optimality gaps in the multilayer context.

We study a planning problem arising in SDH/WDM multi-layer telecommunication network design. The goal is to find a minimum cost installation of link and node hardware of both network layers such that traffic demands can be realized via grooming and a survivable routing. We present a mixed-integer programming formulation for a predefined set of admissible logical links that takes many practical side constraints into account, including node hardware, several bit-rates, and survivability against single physical node or link failures. This model is solved using a branch-and-cut approach with cutting planes based on either of the two layers. On several realistic two-layer planning scenarios, we show that these cutting planes are still useful in the multi-layer context, helping to increase the dual bound and to reduce the optimality gaps.