We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.

However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.

We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v.