Bicriteria approximation tradeoff for the node-cost budget problem

We consider an optimization problem consisting of an undirected graph, with cost and profit functions defined on all vertices. The goal is to find a connected subset of vertices with maximum total profit, whose total cost does not exceed a given budget. The best result known prior to this work guaranteed a (2, O(log n)) bicriteria approximation that is, the solution's profit is at least a fraction of 1 O(log n) of an optimum solution respecting the budget, while its cost is at most twice the given budget. We improve these results and present a bicriteria tradeoff that, given any e ? (0, 1], guarantees a (1 + e, O( 1 e log n))-approximation.