We consider the problem of finding efficient trees to send information from k sources to a single sink in a network where information can be aggregated at intermediate nodes in the tree. Specifically, we assume that if information from j sources is traveling over a link, the total information that needs to be transmitted is f(j). One natural and important (though not necessarily comprehensive) class of functions is those which are concave, non-decreasing, and satisfy f(0) = 0. Our goal is to find a tree which is a good approximation simultaneously to the optimum trees for all such functions. This problem is motivated by aggregation in sensor networks, as well as by buy-at-bulk network design.

We give simple and easy-to-analyze randomized approximation algorithms for several well-studied NP-hard network design problems. Our algorithms improve over the previously best known approximation ratios. Our main results are the following. We give a randomized 3.55-approximation algorithm for the connected facility location problem. The algorithm requires three lines to state, one page to analyze, and improves the best-known performance guarantee for the problem. We give a 5.55-approximation algorithm for virtual private network design. Previously, constant-factor approximation algorithms were known only for special cases of this problem. We give a simple constant-factor approximation algorithm for the single-sink buy-at-bulk network design problem. Our performance guarantee improves over what was previously known, and is an order of magnitude improvement over previous combinatorial approximation algorithms for the problem.

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We present the first constant approximation to the single sink buy-at-bulk network design problem, where we have to design a network by buying pipes of different costs and capacities per unit length to route demands at a set of sources to a single sink. The distances in the underlying network form a metric. This result improves the previous bound of O(log |R|), where R is the set of sources. We also present a better constant approximation to the related Access Network Design problem. Our algorithms are randomized and combinatorial. As a subroutine in our algorithm, we use an interesting variant of facility location with lower bounds on the amount of demand an open facility needs to serve. We call this variant load balanced facility location, and present a constant factor approximation for it, while relaxing the lower bounds by a constant factor.

We consider the Buy-at-Bulk network design problem in which we wish to design a network for carrying multicommodity demands from a set of source nodes to a set of destination nodes. The key feature of the problem is that the cost of capacity on each edge is concave and hence exhibits economies of scale. If the cost of capacity per unit length can be different on different edges then we say that the problem is non-uniform. The problem is uniform otherwise.

We study the multicommodity rent-or-buy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can be rented, with cost incurred on a per-unit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. The first constant-factor approximation algorithm for this problem was recently given by Kumar et al. (FOCS ’02); however, this algorithm and its analysis are both quite complicated, and its performance guarantee is extremely large. In this paper, we give a conceptually simple 12approximation algorithm for this problem. Our analysis of this algorithm makes crucial use of cost sharing, the task of allocating the cost of an object to many users of the object in a “fair ” manner. While techniques from approximation algorithms have recently yielded new progress on cost sharing problems, our work is the first to show the converse— that ideas from cost sharing can be fruitfully applied in the design and analysis of approximation algorithms. 1

Today, an increasing number of important network services, such as content distribution, replicated services, and storage systems, are deploying overlays across multiple Internet sites to deliver better performance, reliability and adaptability. Currently however, such network services must individually reimplement substantially similar functionality. For example, applications must configure the overlay to meet their specific demands for scale, service quality and reliability. Further, they must dynamically map data and functions onto network resources---including servers, storage, and network paths---to adapt to changes in load or network conditions.

Abstract. Buy-at-bulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multi-commodity flow demand between node pairs. We present approximation algorithms for buy-at-bulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first poly-logarithmic approximation ratio for the nonuniform problem that allows different cost functions on each edge and node; the ratio we achieve is O(log4 h) where h is the number of demand pairs. In addition we present an O(log h) approximation for the single sink problem. Poly-logarithmic ratios for some related problems are also obtained. Our algorithm for the multi-commodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems. Key words. Non-uniform buy-at-bulk, network design, approximation algorithm, concave cost, network flow, economies of scale AMS subject classifications. 68Q25, 68W25, 90C27, 90C59 1. Introduction. Network

The COST-DISTANCE network design problem is the following. We are given an undirected graph, a designated root vertex, and a set of terminals. We are also given two non-negative real valued functions defined on, namely, a cost function and a length function, and a non-negative weight function on the set. The goal is to find a tree that connects the terminals in to the root and minimizes!"$ # , where is the length of the path in from + to. We give a deterministic. 2435,/10 approximation algorithm for the COST-DISTANCE network design problem, in a sense derandomizing the algorithm given in [4]. Our algorithm is based on a natural 6/70)2835 linear programming relaxation of the problem and in the process we show that its integrality. gap is. Introduction: We study the COST-DISTANCE network design problem, recently defined by Meyerson,

Consider the following network design problem: given a network G = (V, E), source-sink pairs {si, ti} arrive and desire to send a unit of flow between themselves. The cost of the routing is this: if edge e carries a total of fe flow (from all the terminal pairs), the cost is given by ∑ e ℓ(fe), where ℓ is some concave cost function; the goal is to minimize the total cost incurred. However, we want the routing to be oblivious: when terminal pair {si, ti} makes its routing decisions, it does not know the current flow on the edges of the network, nor the identity of the other pairs in the system. Moreover, it does not even know the identity of the function ℓ, merely knowing that ℓ is a concave function of the total flow on the edge. How should it (obliviously) route its one unit of flow?