Upgrading Shortest Paths in Networks

We introduce the Upgrading Shortest Paths Problem, a new combinatorial problem for improving network connectivity with a wide range of applications from multicast communication to wildlife habitat conservation. We define the problem in terms of a network with node delays and a set of node upgrade actions, each associated with a cost and an upgraded (reduced) node delay. The goal is to choose a set of upgrade actions to minimize the shortest delay paths between demand pairs of terminals in the network, subject to a budget constraint. We show that this problem is NP-hard. We describe and test two greedy algorithms against an exact algorithm on synthetic data and on a real-world instance from wildlife habitat conservation. While the greedy algorithms can do arbitrarily poorly in the worst case, they perform fairly well in practice. For most of the instances, taking the better of the two greedy solutions accomplishes within 5% of optimal on our benchmarks.