Node-Optimal Connected k-Subgraphs

In this paper we consider the problem of optimally selecting a set of k nodes in a node-weighted graph, under the requirement that the subgraph induced by the set of nodes selected is connected. This problem arises in a Norwegian oo-shore oil-drilling application. We show this problem to be NP-complete on general graphs, but provide a polynomial time algorithm to nd an optimal solution for trees. We conclude with some approximation algorithms for the problem on general graphs. 0 Introduction Consider the Connected k-Subgraph problem, deened on a graph whose nodes have non-negative weights, in which the objective is to nd a connected subgraph on k nodes such that the sum of the weights of the nodes selected is minimized (or, maximized). This problem models a situation that arises in a Norwegian oo-shore oil-drilling application. Given a set of locations where a facility can be set up, each with a certain cost or beneet, the problem is to select (up to) k contiguous facilities to build, such that the overall beneet is maximized. By modeling the facilities as nodes, with the node weight corresponding to the cost or beneet of the associated facility, and the connectivity of two stations represented by an edge in the graph, we can see that the oil-drilling optimization problem can be modeled by the Connected k-Subgraph problem. Another problem that can be formulated by Connected k-Subgraph comes from forest harvesting: imagine the problem of optimally selecting k cells to harvest, where each cell has an associated beneet, such that the harvested cells must be contiguous. In Section 1, we show that both the maximization and minimization Connected k-Subgraph problems are NP-hard, even if the input graph is assumed to be either bipartite or planar, and