Handling software upgradeability problems with MILP solvers

Upgradeability problems are a critical issue in modern operating systems. The problem consists in finding the "best" solution according to some criteria, to install, remove or upgrade packages in a given installation. This is a difficult problem: the complexity of the upgradeability problem is NP complete and modern OS contain a huge number of packages (often more than 20 000 packages in a Linux distribution). Moreover, several optimisation criteria have to be considered, e.g., stability, memory efficiency, network efficiency. In this paper we investigate the capabilities of MILP solvers to handle this problem. We show that MILP solvers are very efficient when the resolution is based on a linear combination of the criteria. Experiments done on real benchmarks show that the best MILP solvers outperform CP solvers and that they are significantly better than Pseudo Boolean solvers.Comment: In Proceedings LoCoCo 2010, arXiv:1007.083

On the Complexity of the Bipartite Polarization Problem: from Neutral to Highly Polarized Discussions

The Bipartite Polarization Problem is an optimization problem where the goal is to find the highest polarized bipartition on a weighted and labelled graph that represents a debate developed through some social network, where nodes represent user's opinions and edges agreement or disagreement between users. This problem can be seen as a generalization of the maxcut problem, and in previous work approximate solutions and exact solutions have been obtained for real instances obtained from Reddit discussions, showing that such real instances seem to be very easy to solve. In this paper, we investigate further the complexity of this problem, by introducing an instance generation model where a single parameter controls the polarization of the instances in such a way that this correlates with the average complexity to solve those instances. The average complexity results we obtain are consistent with our hypothesis: the higher the polarization of the instance, the easier is to find the corresponding polarized bipartition