Connections in Networks: Hardness of Feasibility versus Optimality

We study the complexity of combinatorial problems that consist of competing infeasibility and optimization components. In particular, we investigate the complexity of the connection subgraph problem, which occurs, e.g., in resource environment economics and social networks. We present results on its worst-case hardness and approximability. We then provide a typical-case analysis by means of a detailed computational study. First, we identify an easy-hard-easy pattern, coinciding with the feasibility phase transition of the problem. Second, our experimental results reveal an interesting interplay between feasibility and optimization. They surprisingly show that proving optimality of the solution of the feasible instances can be substantially easier than proving infeasibility of the infeasible instances in a computationally hard region of the problem space. We also observe an intriguing easy-hard-easy profile for the optimization component itself.