Kontsevich and Batalin–Vilkovisky classes in the ribbon graph complex

The ribbon graph complex computes the cohomology of moduli spaces of complex algebraic curves of all genera and with any number of marked points.
An A-infinity algebra with an even invariant scalar product determines a cycle in the graph complex. Similarly, a differential contractible algebra with an odd invariant scalar product determines a cocycle in the graph complex. These constructions were originally sketched by Kontsevich over a decade ago but until now they have not been properly utilized.
Using homologucal algebra and a finite dimensional analogue of the Batalin–Vilkovisky formalism in quantum field theory we give a conceptual reformulation of these constructions. We also construct infinite series of explicit nontrivial examples of these classes and compute their pairings in terms of asymptotic expansions of certain finite-dimensional integrals.