Design networks with bounded pairwise distance

Abstract

We study the following network design problem: Given a communication network, find a minimum cost subset of missing links such that adding these links to the network makes every pair of points within distance at most d from each other. The problem has been studied earlier [17] under the assumption that all link costs as well as link lengths are identical, and was shown to be R(logn)-hard for every d 2 4. We present a novel linear programming based approach to obtain an O(log la log d) approximation algorithm for the case of uniform link lengths and costs. We also extend the Cl(Iogn) hardness to d E {Z, 3). On the other hand, if link costs can vary, we show that the prob” ‘-’ n lem is n(Z s )hard for d > 3. This version of our problem can be viewed as a special case of the minimum cost d-spanner problem and thus our hardness result applies there as well. For d = 2, however, we show that the problem continues to be O(logn) approximable by giving an O(log n)-approximation to the more general minimum cost Z-spanner problem. An n(2”s’-’ “)-hardness result also holds when all link costs are identical but link lengths may vary (applies even when all lengths are 1 or 2). Our reduction from the label cower problem [3] also applies to another well-studied network design problem. We show that the directed genemlized steiner network problem [6] is n(2 I’&-’ “)-hard, significantly improving upon the Q(logn) hardness known prior to our work. We also present O(n log d) approximation algorithm for our problem under arbitrary link costs and polynomially bounded link lengths. Same result holds for the minimum cost d-spanner problem. Finally, all our positive results extend to the case where each pair (u,u) of nodes has a distinct distance requirement, say d(u, v). The approximation guarantees above hold provided d is replaced by max,,, d(u, v). All our algorithmic as well as hardness results hold for both undirected and directed versions of the problem. Sanjeev Khanna