Theory of Computing Systems

, Volume 53, Issue 2, pp 341–363 | Cite as

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

  • Petr Kolman
  • Christian Scheideler


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 kf) 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.


Multicommodity flow Approximation algorithms Duality 



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


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic
  2. 2.Dept. of Computer ScienceUniversity of PaderbornPaderbornGermany

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