We study a class of robust network design problems motivated by the need to scale core networks to meet increasingly dynamic capacity demands. Past work has focused on designing the network to support all hose matrices (all matrices not exceeding marginal bounds at the nodes). This model may be too conservative if additional information on traffic patterns is available. Another extreme is the fixed demand model, where one designs the network to support peak point-to-point demands. We introduce a capped hose model to explore a broader range of traffic matrices which includes the above two as special cases. It is known that optimal designs for the hose model are always determined by single-hub routing, and for the fixed- demand model are based on shortest-path routing. We shed light on the wider space of capped hose matrices in order to see which traffic models are more shortest path-like as opposed to hub-like. To address the space in between, we use hierarchical multi-hub routing templates, a generalization of hub and tree routing. In particular, we show that by adding peak capacities into the hose model, the single-hub tree-routing template is no longer cost-effective. This initiates the study of a class of robust network design (RND) problems restricted to these templates. Our empirical analysis is based on a heuristic for this new hierarchical RND problem. We also propose that it is possible to define a routing indicator that accounts for the strengths of the marginals and peak demands and use this information to choose the appropriate routing template. We benchmark our approach against other well-known routing templates, using representative carrier networks and a variety of different capped hose traffic demands, parameterized by the relative importance of their marginals as opposed to their point-to-point peak demands.
The present work's focus is not on congestion, but on total link capacity cost. This falls in the general space of @math problems, which includes the Steiner tree and VPN problems as special cases. Exact or constant-approximation polytime algorithms for several other important traffic models have also been designed, e.g. the so-called symmetric and asymmetric hose models @cite_22 @cite_12 . The latter is however @math -hard to approximate in undirected graphs within polylogarithmic factors for general polytopes @cite_12 .
@cite_22 @cite_12
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We consider the following network design problem. We are given an undirected graph G=(V,E) with edges costs c(e) and a set of terminal nodes W. A hose demand matrix for W is any symmetric matrix [Dij] such that for each i, ∑ 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. and 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 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 [Hurkens 2005, Grandoni 2007, Fiorini 2007]. Our main theorem establishes that this conjecture is true in general graphs. This also gives the first polynomial time algorithm for the single path VPN problem. We also show that the multipath version of the conjecture is false. We consider robust (undirected) network design (RND) problems where the set of feasible demands may be given by an arbitrary convex body. This model, introduced by Ben-Ameur and Kerivin [Ben-Ameur W, Kerivin H (2003) New economical virtual private networks. Comm. ACM 46(6):69–73], generalizes the well-studied virtual private network (VPN) problem. Most research in this area has focused on constant factor approximations for specific polytope of demands, such as the class of hose matrices used in the definition of VPN. As pointed out in Chekuri [Chekuri C (2007) Routing and network design with robustness to changing or uncertain traffic demands. SIGACT News 38(3):106–128], however, the general problem was only known to be APX-hard (based on a reduction from the Steiner tree problem). We show that the general robust design is hard to approximate to within polylogarithmic factors. We establish this by showing a general reduction of buy-at-bulk network design to the robust network design problem. Gupta pointed... Abstract This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum-length routing. More specifically, it is shown that in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is Θ(1), whereas in a Euclidean ball, the same proportion scales as Θ(1 R n−1). This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as volume2(B), whereas the same traffic load scales as volume1+1 n (B) in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel database and synthetic ... We present a short geometric proof for the price of anarchy results that have recently been established in a series of papers on selfish routing in multicommodity flow networks. This novel proof also facilitates two new types of results: On the one hand, we give pseudo-approximation results that depend on the class of allowable cost functions. On the other hand, we derive improved bounds on the inefficiency of Nash equilibria for situations in which the equilibrium travel times are within reasonable limits of the free-flow travel times. These tighter bounds help to explain empirical observations in vehicular traffic networks. Our analysis holds in the more general context of congestion games, which provides the framework in which we describe this work.