Abstract
An elementary hroute flow, for an integer h >= 1, is a set of h edgedisjoint paths between a source and a sink, each path carrying a unit of flow, and an hroute flow is a nonnegative linear combination of elementary hroute flows. An hroute cut is a set of edges whose removal decreases the maximum hroute flow between a given sourcesink pair (or between every sourcesink 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 hroute cut is at least f/h and at most O(log^3(k)f) where f is the size of the maximum hroute flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomialtime approximation algorithm for the minimum hroute 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 ballgrowing. 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.
BibTeX  Entry
@InProceedings{kolman_et_al:LIPIcs:2011:3005,
author = {Petr Kolman and Christian Scheideler},
title = {{Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel BallGrowing}},
booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
pages = {129140},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897255},
ISSN = {18688969},
year = {2011},
volume = {9},
editor = {Thomas Schwentick and Christoph D{\"u}rr},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3005},
URN = {urn:nbn:de:0030drops30051},
doi = {10.4230/LIPIcs.STACS.2011.129},
annote = {Keywords: Multicommodity flow, Multiroute flow, Cuts, Duality}
}
Keywords: 

Multicommodity flow, Multiroute flow, Cuts, Duality 
Seminar: 

28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) 
Issue Date: 

2011 
Date of publication: 

11.03.2011 