  We consider two formulations of the random-link fractional matching problem,
a relaxed version of the more standard random-link (integer) matching problem.
In one formulation, we allow each node to be linked to itself in the optimal
matching configuration. In the other one, on the contrary, such a link is
forbidden. Both problems have the same asymptotic average optimal cost of the
random-link matching problem on the complete graph. Using a replica approach
and previous results of W\"{a}stlund [Acta Mathematica 204, 91-150 (2010)], we
analytically derive the finite-size corrections to the asymptotic optimal cost.
We compare our results with numerical simulations and we discuss the main
differences between random-link fractional matching problems and the
random-link matching problem.
