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-bounded 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-constrained network design problems. A cornerstone for this development was the work of Singh and Lau, who showed that for the degree-bounded MST problem, one can 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, where several linear constraints are imposed on the edges adjacent to a vertex v. 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 at v by a constant, instead of violating the whole family of constraints at v by at most a constant number of edges. Furthermore, previous iterative rounding approaches are not suited for degree constraints where some edges are in a super-constant number of constraints. We extend iterative rounding/relaxation approaches, both conceptually as well as in their analysis, to address these limitations. Based on these extensions, we present an 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 of cost at most OPT, such that for every vertex v, it suffices to remove at most 8 edges from the spanning tree to satisfy the matroidal degree constraint at v.
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