The VPN Conjecture Is True

We consider the following network design problem. We are given an undirected graph <i>G</i> = (<i>V</i>,<i>E</i>) with edge costs <i>c</i>(<i>e</i>) and a set of terminal nodes <i>W</i> ⊆ <i>V</i>. A <i>hose</i> demand matrix is any symmetric matrix <i>D</i>, indexed by the terminals, such that for each <i>i</i> ∈ <i>W</i>, ∑<sub>j≠i</sub> <i>D</i><sub>ij</sub> ≤ 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 <i>oblivious routing</i> template, in this context, is a simple path <i>P</i><sub>ij</sub> for each pair <i>i,j</i> ∈ <i>W</i>. Given such a template, if we are to route a demand matrix <i>D</i>, then for each <i>i,j</i>, we send <i>D</i><sub>ij</sub> units of flow along each <i>P</i><sub>ij</sub>. 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 <i>VPN Conjecture</i>. 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].

[1]  Leslie G. Valiant,et al.  A Scheme for Fast Parallel Communication , 1982, SIAM J. Comput..

[2]  William J. Cook,et al.  Combinatorial Optimization: Cook/Combinatorial , 1997 .

[3]  Georgios Petrou,et al.  Robust network design in telecommunications under polytope demand uncertainty , 2010, Eur. J. Oper. Res..

[4]  Edith Cohen,et al.  Optimal oblivious routing in polynomial time , 2004, J. Comput. Syst. Sci..

[5]  Gianpaolo Oriolo,et al.  The VPN Problem with Concave Costs , 2010, SIAM J. Discret. Math..

[6]  Walid Ben-Ameur,et al.  NEW ECONOMICAL VIRTUAL PRIVATE NETWORKS , 2003 .

[7]  Edoardo Amaldi,et al.  Provisioning virtual private networks under traffic uncertainty , 2007, Networks.

[8]  Fabrizio Grandoni,et al.  Network Design via Core Detouring for Problems without a Core , 2010, ICALP.

[9]  Friedrich Eisenbrand,et al.  An improved approximation algorithm for virtual private network design , 2005, SODA '05.

[10]  Anwar Elwalid,et al.  Exploiting parallelism to boost data-path rate in high-speed IP/MPLS networking , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

[11]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[12]  Amit Kumar,et al.  Provisioning a virtual private network: a network design problem for multicommodity flow , 2001, STOC '01.

[13]  Edith Cohen,et al.  Making intra-domain routing robust to changing and uncertain traffic demands: understanding fundamental tradeoffs , 2003, SIGCOMM '03.

[14]  Thomas Erlebach,et al.  Optimal bandwidth reservation in hose-model VPNs with multi-path routing , 2004, IEEE INFOCOM 2004.

[15]  Amit Kumar,et al.  Algorithms for provisioning virtual private networks in the hose model , 2002, TNET.

[16]  Martin Skutella,et al.  New Approaches for Virtual Private Network Design , 2005, SIAM J. Comput..

[17]  Walid Ben-Ameur,et al.  Routing of Uncertain Traffic Demands , 2005 .

[18]  Peter J. Winzer,et al.  Robust network design and selective randomized load balancing , 2005 .

[19]  Gianpaolo Oriolo,et al.  Hardness of robust network design , 2007 .

[20]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[21]  Debasis Mitra,et al.  Randomized parallel communications on an extension of the omega network , 1987, JACM.

[22]  N. McKeown,et al.  Designing a Predictable Internet Backbone Network , 2004 .

[23]  Friedrich Eisenbrand,et al.  Provisioning a Virtual Private Network Under the Presence of Non-communicating Groups , 2006, CIAC.

[24]  A Gerodimos,et al.  Robust Discrete Optimization and its Applications , 1996, J. Oper. Res. Soc..

[25]  Allan Borodin,et al.  Routing, Merging, and Sorting on Parallel Models of Computation , 1985, J. Comput. Syst. Sci..

[26]  Subhash Suri,et al.  Designing Least-Cost Nonblocking Broadband Networks , 1997, J. Algorithms.

[27]  Laurent El Ghaoui,et al.  Robust Solutions to Least-Squares Problems with Uncertain Data , 1997, SIAM J. Matrix Anal. Appl..

[28]  Sudipta Sengupta,et al.  Efficient and robust routing of highly variable traffic , 2005 .

[29]  Albert G. Greenberg,et al.  A flexible model for resource management in virtual private networks , 1999, SIGCOMM '99.

[30]  Frank Thomson Leighton,et al.  Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms , 1999, JACM.

[31]  F. B. Shepherd,et al.  Selective randomized load balancing and mesh networks with changing demands , 2006 .

[32]  Chandra Chekuri.  Routing and network design with robustness to changing or uncertain traffic demands , 2007, SIGA.

[33]  Panos Kouvelis,et al.  Approaches for Handling Uncertainty in Decision Making , 1997 .

[34]  Laurent El Ghaoui,et al.  Robust Solutions to Uncertain Semidefinite Programs , 1998, SIAM J. Optim..

[35]  Tim Roughgarden,et al.  Simpler and better approximation algorithms for network design , 2003, STOC '03.

[36]  Jean-Philippe Vial,et al.  Robust Optimization , 2021, ICORES.

[37]  Melvyn Sim,et al.  Robust discrete optimization and network flows , 2003, Math. Program..

[38]  Gianpaolo Oriolo,et al.  The VPN Tree Routing Conjecture for Outerplanar Networks , 2007 .

[39]  Arkadi Nemirovski,et al.  Robust solutions of uncertain linear programs , 1999, Oper. Res. Lett..

[40]  Edoardo Amaldi,et al.  Provisioning virtual private networks under traffic uncertainty , 2007 .

[41]  Vijay P. Kumar,et al.  Switched optical backbone for cost-effective scalable core IP networks , 2003, IEEE Commun. Mag..

[42]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[43]  Judith Keijsper,et al.  Virtual Private Network Design: A Proof of the Tree Routing Conjecture on Ring Networks , 2007, SIAM J. Discret. Math..

[44]  William J. Cook,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[45]  Allan Borodin,et al.  Routing, merging and sorting on parallel models of computation , 1982, STOC '82.

[46]  Cheng-Shang Chang,et al.  Load balanced Birkhoff-von Neumann switches, part I: one-stage buffering , 2002, Computer Communications.

[47]  Walid Ben-Ameur,et al.  Networks new economical virtual private , 2003, CACM.

[48]  Harald Räcke,et al.  Minimizing Congestion in General Networks , 2002, FOCS.

[49]  Leslie G. Valiant,et al.  Universal schemes for parallel communication , 1981, STOC '81.

[50]  Laura Sanità,et al.  On the Complexity of the Asymmetric VPN Problem , 2009, APPROX-RANDOM.

[51]  Martin Skutella,et al.  A short proof of the VPN Tree Routing Conjecture on ring networks , 2008, Oper. Res. Lett..

[52]  Harald Räcke,et al.  Optimal hierarchical decompositions for congestion minimization in networks , 2008, STOC.

[53]  Neil Olver,et al.  Approximability of robust network design , 2010, SODA '10.

[54]  Gianpaolo Oriolo,et al.  Design of trees in the hose model: The balanced case , 2006, Oper. Res. Lett..

[55]  Nick McKeown,et al.  Scaling internet routers using optics , 2003, SIGCOMM '03.

[56]  Christos Kaklamanis,et al.  Tight bounds for oblivious routing in the hypercube , 1990, SPAA '90.