Spider Covers for Prize-Collecting Network Activation Problem

In the network activation problem, each edge in a graph is associated with an activation function that decides whether the edge is activated from weights assigned to its end nodes. The feasible solutions of the problem are node weights such that the activated edges form graphs of required connectivity, and the objective is to find a feasible solution minimizing its total weight. In this article, we consider a prize-collecting version of the network activation problem and present the first nontrivial approximation algorithms. Our algorithms are based on a new linear programming relaxation of the problem. They round optimal solutions for the relaxation by repeatedly computing node weights activating subgraphs, called spiders, which are known to be useful for approximating the network activation problem. For the problem with node-connectivity requirements, we also present a new potential function on uncrossable biset families and use it to analyze our algorithms.