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 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.

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

We demonstrate a new connection between fixed-parameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of so-called “bidimensional” problems to show that essentially all such problems have both subexponential fixed-parameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal problems, dominating set, edge dominating set, r-dominating set, diameter, connected dominating set, connected edge dominating set, and connected r-dominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two well-known problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the Lipton-Tarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.

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?

We study the multicommodity buy-at-bulk network design problem in which we seek to design a network that satisfies the demands between terminals from a given set of sourcesink pairs. The key characteristic of this problem is the fact that the cost functions associated with the edges of the graph are sub-additive monotone and hence experience economies of scale. In the non-uniform case, each edge has its own cost function – possibly different from other edges. Special cases of this problem have been studied extensively: there are approximation algorithms when the edge cost functions are identical or when all source-sink pairs share the same source. We present the first non-trivial approximation algorithm for the general case. Our algorithm is an extremely simple randomized greedy algorithm and has an approximation guarantee of exp(O ( √ ln nln ln n)) when the instance has at most n source-sink pairs with unit demands. In the case of general demands, this yields an approximation factor of exp(O ( √ ln nln ln n)) log D, where D is the value of the largest demand.

Abstract not found

Abstract not found

We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00 and for the single-sink rent-or-buy problem from 3.55 to 2.92. We show that our connected facility location algorithms can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems.

We present a very simple way of derandomizing the algorithm proposed by Gupta, Kumar and Roughgarden for Single Source Rent-or-Buy by using the method of conditional expectation. Using the improved analysis of Eisenbrand, Grandoni and Rothvoß, our derandomized algorithm has an approximation guarantee of 3.28. Keywords: Approximation algorithms, Derandomization, Network Design, Rent-or-Buy. 1