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 $\mathcal{NP}$-hard to approximate within a factor of at least 1.1377 for L ≥5 in the case of node-cuts and for L ≥4 in the case of edge-cuts. We also give approximation algorithms of ratio min {L,n/L} in the node case and $\min\{L,n^2/L^2,\sqrt{m}\}$ in the edge case, where n denotes the number of nodes and m denotes the number of edges. We discuss the integrality gaps of the LP relaxations of length-bounded flow and cut problems, analyze the structure of optimal solutions, and present further complexity results for special cases.