Abstract
An elementary h-route flow, for an integer h≥1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h≤3: The size of a minimum h-route cut is at least f/h and at most O(log4 k⋅f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h=3 that has an approximation ratio of O(log4 k). Previously, polylogarithmic approximation was known only for h-route cuts for h≤2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.
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Notes
To be precise, the entry edges of H(r) are a subset of δ(r)∖δ 2(r), and the entry nodes of H(r) are a subset of Z: it may happen that (some of) the nodes in Z are not connected to an outside node (see the definition of entry nodes). However, such a pathological case makes the situation only easier and thus, without loss of generality, we assume that all nodes in Z are the entry nodes of H(r).
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Acknowledgements
The first author would like to thank Jiří Sgall and Thomas Erlebach for stimulating discussions.
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P. Kolman was supported by the Center of Excellence—Institute for Theoretical Computer Science, Prague, project P202/12/G061 of GA ČR. C. Scheideler was supported by DFG SCHE 1592/1-1.
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Kolman, P., Scheideler, C. Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing. Theory Comput Syst 53, 341–363 (2013). https://doi.org/10.1007/s00224-013-9454-3
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DOI: https://doi.org/10.1007/s00224-013-9454-3