Domination between traffic matrices

A traffic matrix D-1 dominates a traffic matrix D-2 if any capacity reservation supporting D-1 supports D-2 as well. We prove that D-1 dominates D-2 if and only if D-1, considered as a capacity reservation, supports D-2. We show several generalizations of this result

Provisioning a virtual private network under the presence of non-communicating groups

Virtual private network design in the hose model deals with the reservation of capacities in a weighted graph such that the terminals in this network can communicate with one another. Each terminal is equipped with an upper bound on the amount of traffic that the terminal can send or receive. The task is to install capacities at minimum cost and to compute paths for each unordered terminal pair such that each valid traffic matrix can be routed along those paths. In this paper we consider a variant of the virtual private network design problem which generalizes the previously studied symmetric and asymmetric case. In our model the terminal set is partitioned into a number of groups, where terminals of each group do not communicate with each other. Our main result is a 4.74 approximation algorithm for this problem. © Springer-Verlag Berlin Heidelberg 2006

On the Complexity of the Asymmetric VPN Problem

We give the first constant factor approximation algorithm for the asymmetric Virtual Private Network (VPN) problem with arbitrary concave costs. We even show the stronger result, that there is always a tree solution of cost at most 2 OPT and that a tree solution of (expected) cost at most 49.84 OPT can be determined in polynomial time. Furthermore, we answer an outstanding open question about the complexity status of the so called balanced VPN problem by proving its NP-hardness

Network Design via Core Detouring for Problems Without a Core

Some of the currently best-known approximation algorithms for network design are based on random sampling. One of the key steps of such algorithms is connecting a set of source nodes to a random subset of them. In a recent work [Eisenbrand,Grandoni,Rothvo\ss,Schäfer-SODA'08], a new technique, \emph{core-detouring}, is described to bound the mentioned connection cost. This is achieved by defining a sub-optimal connection scheme, where paths are detoured through a proper connected subgraph (core). The cost of the detoured paths is bounded against the cost of the core and of the distances from the sources to the core. The analysis then boils down to proving the \emph{existence} of a convenient core. For some problems, such as connected facility location and single-sink rent-or-buy, the choice of the core is obvious (i.e., the Steiner tree in the optimum solution). Other, more complex network design problems do not exhibit any such core. In this paper we show that core-detouring can be nonetheless successfully applied. The basic idea is constructing a convenient core by manipulating the optimal solution in a proper (not necessarily trivial) way. We illustrate that by presenting improved approximation algorithms for two well-studied problems: virtual private network design and single-sink buy-at-bulk

New approaches for virtual private network design

Virtual private network design is the following NP-hard problem. We are given a communication network represented as a weighted graph with thresholds on the nodes which represent the amount of flow that a node can send to and receive from the network. The task is to reserve capacities at minimum cost and to specify paths between every ordered pair of nodes such that all valid traffic-matrices can be routed along the corresponding paths. Recently, this network design problem has received considerable attention in the literature. It is motivated by the fact that the exact amount of flow which is exchanged between terminals is not known in advance and prediction is often elusive. The main contributions of this paper are as follows: (1) Using Hu's 2-commodity flow theorem, we provide a new and considerably stronger lower bound on the cost of an optimum solution. With this lower bound we reanalyze a simple routing scheme which has been described in the literature many times, and provide an improved upper bound on its approximation ratio. (2) We present a new randomized approximation algorithm. In contrast to earlier approaches from the literature, the resulting solution does not have tree structure. A combination of our new algorithm with the simple routing scheme yields an expected performance ratio of 3.79 for virtual private network design. This is a considerable improvement of the previously best known 5.55-approximation result [ A. Gupta, A. Kumar, and T. Roughgarden, Simpler and better approximation algorithms for network design, in Proceedings of the ACM Symposium on Theory of Computing, ACM, New York, 2003, pp. 365 -372]. (3) Our VPND algorithm uses a Steiner tree approximation algorithm as a subroutine. It is known that an optimum Steiner tree can be computed in polynomial time if the number of terminals is logarithmic. Replacing the approximate Steiner tree computation with an exact one whenever the number of terminals is sufficiently small, we finally reduce the approximation ratio to 3.55. To the best of our knowledge, this is the first time that a nontrivial result from exact (exponential) algorithms leads to an improved polynomial-time approximation algorithm