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. The first research on path related constraints we are aware of was done in 1978 by Lovász, Neumann Lara, and Plummer. Among others they show that the minimum length-bounded node-cut problem is polynomial for L≤4. We show that the minimum length-bounded cut problem turns out to be 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 (both in graphs with unit edge lengths). We also give approximation algorithms of ratio min{L,n/L} in the node case and min{L,n2/L2,√m} in the edge case, where n denotes the number of nodes and m denotes the number of edges. For classes of graphs such as constant degree expanders, hypercubes, and butterflies, we obtain O(log n)-approximations by using the Shortening Lemma for flow numbers [Kolman and Scheideler, SODA'02]. We discuss the integrality gaps of the LP relaxations of length-bounded flow and cut problems, which can be of size Ω(√n). We show that edge- and path-flows are not polynomially equivalent for length-bounded flows, analyze the structure of optimal solutions and give instances where each maximum flow ships a large percentage of the flow along paths with an arbitrarily small flow value