Theory of Computing Systems

, Volume 53, Issue 2, pp 341–363

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

Article

## Abstract

An elementaryh-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(log4kf) 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(log4k). 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

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