Length-bounded cuts and flows

For a given number <i>L</i>, an <i>L</i>-length-bounded edge-cut (node-cut, respectively) in a graph <i>G</i> with source <i>s</i> and sink <i>t</i> is a set <i>C</i> of edges (nodes, respectively) such that no <i>s</i>-<i>t</i>-path of length at most <i>L</i> remains in the graph after removing the edges (nodes, respectively) in <i>C</i>. An <i>L</i>-length-bounded flow is a flow that can be decomposed into flow paths of length at most <i>L</i>. In contrast to classical flow theory, we describe instances for which the minimum <i>L</i>-length-bounded edge-cut (node-cut, respectively) is Θ(<i>n</i><sup>2/3</sup>)-times (Θ(&sqrt;<i>n</i>)-times, respectively) larger than the maximum <i>L</i>-length-bounded flow, where <i>n</i> denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is <i>NP</i>-hard to approximate within a factor of 1.1377 for <i>L</i>≥ 5 in the case of node-cuts and for <i>L</i>≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio <i>O</i>(min{<i>L</i>,<i>n/L</i>}) ⊆ <i>O</i>&sqrt;<i>n</i> in the node case and <i>O</i>(min {<i>L</i>,<i>n</i><sup>2</sup>/<i>L</i><sup>2</sup>,&sqrt;<i>m</i>} ⊆ <i>O</i><sup>2/3</sup> in the edge case, where <i>m</i> denotes the number of edges. Concerning <i>L</i>-length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is <i>NP</i>-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.

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