Easy and hard separation of sparse and dense odd-set constraints in matching

We investigate polytopes intermediate between the fractional matching and the perfect matching polytopes, by imposing a strict subset of the odd-set (blossom) constraints. For sparse constraints, we give a polynomial time separation algorithm if only constraints on all odd sets bounded by a given size (e.g. $\le 9+\left|V\right|/6$) are present. Our algorithm also solves the more general problem of finding a T-cut subject to upper bounds on the cardinality of its defining node set and on its cost. In contrast, regarding dense constraints, we prove that for every $0<\alpha \le \frac{1}{2}$, it is NP-complete to separate over the class of constraints on odd sets of size $2\lfloor (1+\alpha \left|V\right|)/2\rfloor -1$ or $\ge \alpha \left|V\right|$.