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We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with link-costs and, for each pair fi; jg of nodes, an edge-connectivity requirement r ij . The goal is to find a minimum-cost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.

We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. 1 Introduction We consider the problem of finding a minimum-cost subgraph of a given graph such that the number of edges crossing each cut is at least a specified requirement. Formally, given an undirected multigraph G = (V; E), a non-negative cost function c : E ! Q+ , and a requirement function f : 2 V ! Z , solve the following integer program (IP): min X e2E c e x...

The primal-dual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primal-dual method can be modified to provide good approximation algorithms for a wide variety of NP-hard problems. We concentrate on results from recent research applying the primal-dual method to problems in network design.

We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent’s goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1 + ε)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65 + ε)-approximate Nash equilibrium that does not cost much more.

The essence of the simplest buy-at-bulk network design problem is buying network capacity "wholesale " to guarantee connectivity from all network nodes to a certain central network switch. Capacity is sold with "volume discount": the more capacity is bought, the cheaper is the price per unit of bandwidth. We provide O(log 2 n) randomized approximation algorithm for the problem. This solves the open problem in [15]. The only previously known solutions were restricted to special cases (Euclidean graphs) [15]. We solve additional natural variations of the problem, such as multi-sink network design, as well as selective network design. These problems can be viewed as generalizations of the the Generalized Steiner Connectivity and Prize-collecting salesman (K-MST) problems. In the selective network design problem, some subset of k wells must be connected to the (single) refinery, so that the total cost is minimized. 1 Introduction 1.1 The basic problem Consider an oil company that wi...

Given a tour visiting n points in a metric space, the latency of one of these points p is the distance traveled in the tour before reaching p. The minimum latency problem asks for a tour passing through n given points for which the total latency of the n points is minimum; in effect, we are seeking the tour with minimum average "arrival time." This problem has been studied in the operations research literature, where it has also been termed the "delivery-man problem" and the "traveling repairman problem." The approximability of the minimum latency problem was first considered by Sahni and Gonzalez in 1976; however, unlike the classical traveling salesman problem, it is not easy to give any constant-factor approximation algorithm for the minimum latency problem. Recently, Blum, Chalasani, Coppersmith, Pulleyblank, Raghavan, and Sudan gave the first such algorithm, obtaining an approximation ratio of 144. In this work, we present an algorithm which improves this ratio to 21:55. The dev...

We give the first approximation algorithm for the node-weighted Steiner tree problem. Its performance guarantee is within a constant factor of the best possible unless ~ P ' NP . Our algorithm generalizes to handle other network design problems.

M.I.T. We present the first polynomial-time approxima-tion algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. If k is the maximum cut requirement of the problem, our solu-tion comes within a factor of 2k of optimal. Our algo-rithm is primal-dual and shows the importance of this technique in designing approximation algorithms. 1

We are given a set of points pl,...,p. and a symmetric distance matrix (o!ij) giving the distance between pi and pj. We wish to construct a tour that minimizes ~~=1 1(z), where l(i) is