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

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## 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^{4} *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^{4} *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 Approximation algorithms Duality## Notes

### Acknowledgements

The first author would like to thank Jiří Sgall and Thomas Erlebach for stimulating discussions.

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