Length-bounded cuts and flows

Abstract

An L-length-bounded cut in a graph G with source s, and sink t is a cut that destroys all s-i-paths of length at most L. An L-length-bounded flow is a flow in which only flow paths of length at most L ave used. We show that the minimum length-bounded cut problem in graphs with unit edge lengths is NP-hard to approximate within a factor of at loaat 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, √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. © Spriner-Verlag Berliu Heidelberg 2006.