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# Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

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LIPIcs.STACS.2011.129.pdf
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## Cite As

Petr Kolman and Christian Scheideler. Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 129-140, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)
https://doi.org/10.4230/LIPIcs.STACS.2011.129

## 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(log^3(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(log^3 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.
##### Keywords
• Multicommodity flow
• Multiroute flow
• Cuts
• Duality

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