Problem Hardness Result Approximation ResultBSTP

Several problems arising in the areas of communication networks and VLSI design can be expressed in the following general form: Enhance the performance of an underlying network by upgrading some of its nodes. We investigate one such problem. Consider a network where nodes represent processors and edges represent bidirectional communication links. The processor at a node v can be upgraded at an expense of cost(v). Such an upgrade reduces the delay of each link emanating from v by a xed factor x, where 0 < x < 1. The goal is to nd a minimum cost set of nodes to be upgraded so that the resulting network has a spanning tree in which each edge is of delay at most a given value. We provide both hardness and approximation results for the problem. We show that the problem is NP-hard and cannot be approximated within any factor < lnn, unless NP DTIME(n log log n), where n is the number of nodes in the network. This result holds even when the cost of upgrading each node is 1. We then present the rst polynomial time approximation algorithms for the problem. For the general case, our approximation algorithm comes within a factor of 2 lnn of the minimum upgrading cost. When the cost of upgrading each node is 1, we present an approximation algorithm with a performance guarantee of 4(2 + ln), where is the maximum node degree. For = o(p n), this algorithm performs better than the general algorithm. In particular, for graphs of bounded node degree, we obtain a constant factor approximation. Finally, we present a polynomial time algorithm for the class of treewidth-bounded graphs.