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The VPN conjecture is true

Goyal, Navin, Olver, Neil ORCID: 0000-0001-8897-5459 and Bruce Shepherd, F. (2013) The VPN conjecture is true. Journal of the ACM, 60 (3). ISSN 0004-5411

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Identification Number: 10.1145/2487241.2487243


We consider the following network design problem. We are given an undirected graph G = (V,E) with edge costs c(e) and a set of terminal nodes W ⊆ V. A hose demand matrix is any symmetric matrix D, indexed by the terminals, such that for each i ∈ W, ∑j≠i Dij ≤ 1. We must compute the minimum-cost edge capacities that are able to support the oblivious routing of every hose matrix in the network. An oblivious routing template, in this context, is a simple path Pij for each pair i,j ∈ W. Given such a template, if we are to route a demand matrix D, then for each i,j, we send Dij units of flow along each Pij. Fingerhut et al. [1997] and Gupta et al. [2001] obtained a 2-approximation for this problem, using a solution template in the form of a tree. It has been widely asked and subsequently conjectured [Italiano et al. 2006] that this solution actually results in the optimal capacity for the single-path VPN design problem; this has become known as the VPN Conjecture. The conjecture has previously been proven for some restricted classes of graphs [Fingerhut et al. 1997; Fiorini et al. 2007; Grandoni et al. 2008; Hurkens et al. 2007]. Our main theorem establishes that this conjecture is true in general graphs. This also has the implication that the single-path VPN problem is solvable in polynomial time. A natural fractional version of the conjecture had also been proposed [Hurkens et al. 2007]. In this version, the routing may split flow between many paths, in specified proportions. We demonstrate that this multipath version of the conjecture is in fact false. The multipath and single path versions of the VPN problem are essentially direct analogues of the randomized and nonrandomized versions of oblivious routing schemes for minimizing congestion for permutation routing [Borodin and Hopcroft 1982; Valiant 1982].

Item Type: Article
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Additional Information: © 2020 ACM
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Jan 2020 10:09
Last Modified: 20 Oct 2021 02:47

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