Enhanced compact models for the connected subgraph problem and for the shortest path problem in digraphs with negative cycles

We investigate the minimum-weight connected subgraph problem. The importance of this problem stems from the fact that it constitutes the backbone of many network design problems having applications in several areas including telecommunication, energy, and distribution planning. We show that this problem is NP-hard, and we propose a new polynomial-size nonlinear mixed-integer programming model. We apply the Reformulation-Linearization Technique (RLT) to linearize the proposed model while keeping a polynomial number of variables and constraints. Furthermore, we show how similar modelling techniques enable an enhanced polynomial size formulation to be derived for the shortest elementary path. This latter problem is known to be intractable and has many applications (in particular, within the context of column generation). We report the results of extensive computational experiments on graphs with up to 1000 nodes. These results attest to the efficacy of the proposed compact formulations. In particular, we show that the proposed formulations consistently outperform compact formulations from the literature.