### Length-bounded cuts and flows

Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Petr Kolman, Ondřej Pangrác, Heiko Schilling, Martin Skutella
2010 ACM Transactions on Algorithms
· Georg Baier et al. 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
more » ... 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.