On definable matchings in o-minimal bipartite graphs

We consider bipartite graphs definable in o-minimal structures, in which the edge relation $G$ is a finite union of graphs of certain measure-preserving maps.

We establish the existence of definable matchings with few short augmenting paths. Under the additional assumptions of $G\subseteq [0,1]^n$ and 2-regularity, this yields the existence of definable matchings covering all vertices outside of a set of arbitrarily small positive measure (Lebesgue measure of the standard part). As an application we obtain an approximate 2-cancellation result for the semigroup of definable subsets of $[0,1]^n$ modulo an equivalence relation induced by measure-preserving maps.