Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing
Résumé
An {\em elementary $h$-route flow}, for an integer $h\geq 1$, is a set of $h$ edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an {\em $h$-route flow} is a non-negative linear combination of elementary $h$-route flows. An {\em $h$-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 $h\leq 3$: The size of a minimum $h$-route cut is at least $f/h$ and at most $O(\log^3 k \cdot f)$ 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 $h\le 2$. 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.
Origine : Accord explicite pour ce dépôt
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