Network improvement problems

Abstract

The authors study budget constrained optimal network improvement problems. Such problems aim at finding optimal strategies for improving a network under some cost measure subject to certain budget constraints. As an example, consider the following prototypical problem: Let G = (V, E) be an undirected graph with two cost values L(e) and C(e) associated with each edge e, where L(e) denotes the length of e and C(e) denotes the cost of reducing the length of e by a unit amount. A reduction strategy specifies for each edge e, the amount by which L(e) is to be reduced. For a given budget B, the goal is to find a reduction strategy such that the total cost of reduction is at most B and the minimum cost tree (with respect to some measure M) under the modified L costs is the best over all possible reduction strategies which obey the budget constraint. Typical measures M for a tree are the total weight and the diameter. They provide both hardness and approximation results for the two measures M mentioned above. For the problem of minimizing the total weight of a spacing tree, they provide an algorithm that, for any fixed {gamma},{var_epsilon} > 0, finds a solution whose weight is at most (1 + 1/{gamma}) times that of a minimum length spanning tree plus an additive constant of at most {var_epsilon} and the total cost of improvement is at most (1 + {gamma}) times the budget B. This result can be extended to obtain approximation algorithms for more general network design problems considered in [GW, GG+94].