We are given an undirected graph G = (V; E) with positive weights on its vertices representing demands, and non-negative costs on its edges. Also given are a capacity constraint k, and root vertex r 2 V . In this paper, we consider the capacitated minimum spanning network (CMSN) problem, which asks for a minimum cost spanning network such that the the removal of r and its incident edges breaks the network into a number of components (groups), each of which is 2-edge-connected with a total weight of at most k. We show that the CMSN problem is NP-hard, and present a 4-approximation algorithm for graphs satisfying triangle inequality. We also show how to obtain similar approximation results for a related 2-vertex-connected CMSN problem.

Abstruct- In third generation (3G) cellular networks, base stations ate connected to base station controllers by point-to-point (usually TlIE1) links. However, today’s TllEl based hackhaul network is not a good match for next generation wireless networks because symmetric Tls is not an efficient way to carry bursty and asymmetric data traffic. In this paper, we propose designing an IEEE 802.16-based wireless radio access network to carry the traffic from the base station to the radio network controller. 802.16 has several characteristics that make it a better match for 36 radio acres % networks including its support for Time Division Duplex mode that supports asymmetry eficiently. In this paper, we tackle the following question: given a layout of base stations and base station controllers, how do we design the topology of the 802.16 radio access network connecting the base stations to the base station controller that minimizes the number of 802.16 links used while meeting the expected demands of traffic frodto the base station?? We make three contributions: we first sbow that finding the optimal solution to the problem is NP-hard, We then provide heuristics that perform close to the optimal solution. Finally, we address the reliability issue of failure of 802.16 links or nodes by designing algorithms to create topologies that can handle single failures effectively. I.

Given a set of terminals, each associated with a positive number denoting the traffic to be routed to a central terminal (root), the Capacitated Minimum Spanning Tree (CMST) problem asks for a minimum spanning tree, spanning all terminals, such that the amount of traffic routed from a subtree, linked to the root by an edge, does not exceed the given capacity constraint k. The CMST problem is NP-complete and has been extensively studied for the past 40 years. Current best heuristics, in terms of cost and computation time (O(n log n)), are due to Esau and Williams [1], and Jothi and Raghavachari [2].