A cavity approach to optimization and inverse dynamical problems

In these two lectures we shall discuss how the cavity approach can be used efficiently to study optimization problems with global (topological) constraints and how the same techniques can be generalized to study inverse problems in irreversible dynamical processes. These two classes of problems are formally very similar: they both require an efficient procedure to trace over all trajectories of either auxiliary variables which enforce global constraints, or directly dynamical variables defining the inverse dynamical problems. We will mention three basic examples, namely the Minimum Steiner Tree problem, the inverse threshold linear dynamical problem, and the patient-zero problem in epidemic cascades. All these examples are root problems in optimization and inference over networks. They appear in many modern applications and in a variety of different contexts. Credit for these results should be shared with A. Braunstein, A. Ramezanpour, F. Altarelli, L. Dall'Asta, I. Biazzo and A. Lage-Castellanos.