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

Petr Kolman, Christian Scheideler
2013 Theory of Computing Systems
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
more » ... modity 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.