### The VPN Conjecture Is TrueThe VPN Conjecture Is True

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 Author Goyal, Navin ♦ Olver, Neil ♦ Shepherd, F Bruce Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2013 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Network design ♦ Oblivious routing ♦ Robust optimization Abstract We consider the following network design problem. We are given an undirected graph $\textit{G}$ = $(\textit{V},\textit{E})$ with edge costs $\textit{c}(\textit{e})$ and a set of terminal nodes $\textit{W}$ ⊆ $\textit{V}.$ A $\textit{hose}$ demand matrix is any symmetric matrix $\textit{D},$ indexed by the terminals, such that for each $\textit{i}$ ∈ $\textit{W},$ $∑_{j≠i}$ $D_{ij}$ ≤ 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 $P_{ij}$ for each pair $\textit{i,j}$ ∈ $\textit{W}.$ Given such a template, if we are to route a demand matrix $\textit{D},$ then for each $\textit{i,j},$ we send $D_{ij}$ units of flow along each $P_{ij}.$ 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]. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2013-06-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 60 Issue Number 3 Page Count 17 Starting Page 1 Ending Page 17

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Source: ACM Digital Library