  We study the random-link matching problem on random regular graphs, alongside
with two relaxed versions of the problem, namely the fractional matching and
the so-called "loopy" fractional matching. We estimated the asymptotic average
optimal cost using the cavity method. Moreover, we also study the finite-size
corrections due to rare topological structures appearing in the graph at large
sizes. We estimate these contributions using the cavity approach, and we
compare our results with the output of numerical simulations. The analysis also
clarifies the meaning of the finite-size contributions appearing in the
fully-connected version of the problem, that has been already analyzed in the
literature.
