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 (undirected) Steiner Network problem is: given a graph G = (V, E) with edge/node weights and connectivity requirements {r(u, v) : u, v ∈ U ⊆ V}, find a minimum weight subgraph H of G containing U so that the uv-edge-connectivity in H is at least r(u, v) for all u, v ∈ U. The seminal paper of Jain [19], and nume-rous papers preceding it, considered the Edge-Weighted Steiner Network problem, with weights on the edges only, and developed novel tools for approximating minimum weight edge-covers of several types of set functions and families. However, for the Node-Weighted Steiner Network (NWSN) problem, nontrivial approximation algorithms were known only for 0, 1 requirements. We make an attempt to change this situation, by giving the first non-trivial approxi-mation algorithm for NWSN with arbitrary requirements. Our approximation ratio for NWSN is rmax · O(ln |U|), where rmax = maxu,v∈U r(u, v). This generalizes the result of Klein and Ravi [21] for the case rmax = 1. We also give an O(ln |U|)-approximation algorithm for the node-connectivity variant of NWSN (when the paths are required to be internally-disjoint) for the case rmax = 2. Our results are based on a much more general approximation algorithm for the problem of finding a minimum node-weighted edge-cover of an uncrossable set-family. We also give the first evidence that a polylogarithmic approximation ratio for NWSN might not exist even for |U | = 2 and unit weights. 1 1

We consider approximation algorithms for buy-at-bulk network design, with the additional constraint that demand pairs be protected against a single edge or node failure in the network. In practice, the most popular model used in high speed telecommunication networks for protection against failures, is the so-called 1+1 model. In this model, two edge or node-disjoint paths are provisioned for each demand pair. We obtain the first nontrivial approximation algorithms for buy-at-bulk network design in the 1+1 model for both edge and nodedisjoint protection requirements. Our results are for the single-cable cost model, which is prevalent in optical networks. More specifically, we present a constant-factor approximation for the single-sink case, and an O ( log 3 n) approximation for the multi-commodity case. These results are of interest for practical applications and also suggest several new challenging theoretical problems.

Abstract Buy-at-bulk network design problems arise in settings where the costs for purchasing or installing equip-ment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given muliti-commodity flow demand between node pairs. We present approximation algorithms for buy-at-bulknetwork 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 non-uniform problem that allows different cost functions oneach edge and node; the ratio we achieve is O(log4 h) where h is the number of demand pairs. In additionwe present an O(log h) approximation for the single sink problem. Poly-logarithmic ratios for some relatedproblems 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 but usefultechnique for other network design problems.

Abstract — We obtain the first online algorithms for the node-weighted Steiner tree, Steiner forest and group Steiner tree problems that achieve a poly-logarithmic competitive ratio. Our algorithm for the Steiner tree problem runs in polynomial time, while those for the other two problems take quasi-polynomial time. Our algorithms can be viewed as online LP rounding algorithms in the framework of Buchbinder and Naor (Foundations and Trends in Theoretical Computer Science, 2009); however, while the natural LP formulation of these problems do lead to fractional algorithms with a polylogarithmic competitive ratio, we are unable to round these LPs online without losing a polynomial factor. Therefore, we design new LP formulations for these problems drawing on a combination of paradigms such as spider decompositions, lowdepth Steiner trees, generalized group Steiner problems, etc. and use the additional structure provided by these to round the more sophisticated LPs losing only a poly-logarithmic factor in the competitive ratio. As further applications of our techniques, we also design polynomial-time online algorithms with poly-logarithmic competitive ratios for two fundamental network design problems in edge-weighted graphs: the group Steiner forest problem (thereby resolving an open question raised by Chekuri et al (SODA 2008)) and the single source ℓ-vertex connectivity problem (which complements similar results for the corresponding edge-connectivity problem due to Gupta et al (STOC 2009)). 1.

This thesis presents approximation algorithms for some N P-Hard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widely-believed complexity-theoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomial-time) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N P-Hard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing fault-tolerant networks. Two vertices u, v in a network are said to be k-edge-connected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are k-vertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning