Abstract
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.
This work was partly supported by the Federal Ministry of Education and Research (BMBF grant 03-MOM4B1), by the European Commission – Fet Open project DELIS IST-001907 (SBF grant 03.0378-1), and by the German Research Foundation (DFG grants MO 446/5-2 and SK 58/5-3).
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Baier, G., Erlebach, T., Hall, A., Köhler, E., Schilling, H., Skutella, M. (2006). Length-Bounded Cuts and Flows. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds) Automata, Languages and Programming. ICALP 2006. Lecture Notes in Computer Science, vol 4051. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11786986_59
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