The Node-Weighted Steiner Problem in Graphs of Restricted Node Weights

In this paper we study a variant of the Node-Weighted Steiner Tree problem in which the weights (costs) of vertices are restricted, in the sense that the ratio of the maximum node weight to the minimum node weight is bounded by a quantity α. This problem has applications in multicast routing where the cost of participating routers must be taken into consideration and the network is relatively homogenous in terms of the cost of the routers. We consider both online and offline versions of the problem. For the offline version we show an upper bound of O(min{log α, log k}) on the approximation ratio of deterministic algorithms (where k is the number of terminals). We also prove that the bound is tight unless P = NP . For the online version we show a tight bound of Θ(max{min{α, k}, log k}), which applies to both deterministic and randomized algorithms. We also show how to apply (and extend to node-weighted graphs) recent work of Alon et al. so as to obtain a randomized online algorithm with competitive ratio O(log m log k), where m is the number of the edges in the graph, independently of the value of α. All our bounds also hold for the Generalized Node-Weighted Steiner Problem, in which only connectivity between pairs of vertices must be guaranteed.