Matroidal Degree-Bounded Minimum Spanning Trees

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.

💡 Research Summary

The paper studies a natural generalization of the classic degree‑bounded Minimum Spanning Tree (MST) problem. Instead of a simple upper bound on the degree of each vertex, the authors allow an arbitrary matroid M_v = (δ(v), I_v) on the set of edges incident to each vertex v. A feasible spanning tree T must satisfy T∩δ(v)∈I_v for every v. This model captures partition constraints (where the incident edges are partitioned and each part has its own quota) as well as laminar constraints, and it can represent situations where a single edge participates in many different degree constraints.

The main contribution is a polynomial‑time algorithm that returns a spanning tree of cost at most OPT (the cost of an optimal tree that respects all matroid constraints) while violating each vertex’s matroid constraint by at most eight edges. In other words, for every vertex v one can delete at most eight incident edges from the output tree and obtain a set that belongs to I_v.

The algorithm follows an iterative rounding / relaxation framework but introduces two novel “degree‑adaptation” steps that differ from previous work. At each iteration the algorithm solves a linear program (LP 1) that combines the spanning‑tree polytope with the matroid polytopes of the current degree constraints. The LP solution x is a basic feasible solution; edges with x(e)=0 are discarded, edges with x(e)=1 are contracted, and a maximal family of linearly independent tight spanning‑tree constraints is fixed.

After this preprocessing the algorithm attempts one of two adaptations:

• Type A adaptation – For a vertex w, let U be the set of incident edges that still belong to both degree constraints (i.e., to the matroid of w and to the matroid of the opposite endpoint). If |U|−x(U) ≤ 4, the whole set U is removed from the degree constraint at w. This eliminates the “double‑counted’’ edges from one side, guaranteeing that each such edge will be removed from at least one endpoint’s constraint in the future.

• Type B adaptation – The algorithm defines a special set Q of vertices that are “tight’’ with respect to the current LP (they have a full unit of fractional degree in the subgraph induced by the yet‑unprocessed vertices). For any vertex w not in Q, if the set U of edges that belong to w’s degree constraint but are not incident to any vertex in Q satisfies |U|−x(U) ≤ 4, then U is removed from w’s constraint. This leverages the fact that vertices in Q are already heavily constrained by spanning‑tree requirements, allowing us to ignore many edges when adapting other vertices.

A key technical insight is a sparsity bound on the support of a basic LP solution. If a vertex v has k linearly independent tight matroid constraints, then the fractional degree x(δ(v)) is at least k. Summing over all vertices yields ∑_v x(δ(v)) = 2(|V|−1) (because x lies in the spanning‑tree polytope), implying that the total number of linearly independent tight matroid constraints across the whole graph is O(|V|). Consequently, at any iteration there must exist a vertex for which one of the two adaptations can be applied; otherwise the average fractional degree would be too high, contradicting the sparsity bound.

The algorithm repeats the solve‑adapt‑contract loop until the graph collapses to a single vertex. The contracted edges are then expanded to obtain a spanning tree T in the original graph. By construction, each adaptation removes at most four fractional units from a vertex’s degree, and each vertex can be involved in at most two adaptations (once as a Type A vertex and once as a Type B vertex). Hence at most eight edges per vertex need to be deleted to restore feasibility with respect to the original matroid M_v.

The cost analysis is straightforward: every LP solution respects the objective value of the original LP, which is a lower bound on OPT. Contracted edges have either zero cost (x=0) or are already part of an optimal fractional solution (x=1), so the final tree’s total cost never exceeds OPT.

Overall, the paper extends iterative rounding techniques to a much richer class of degree constraints, overcoming two major limitations of prior work: (i) the inability to keep the total violation bounded when many constraints share edges, and (ii) the reliance on each edge appearing in only a constant number of constraints. By using matroid polytope structure and a two‑phase adaptation scheme, the authors achieve a constant‑factor violation (8) for arbitrary matroidal degree bounds while preserving optimal cost. This result opens the door to practical network‑design applications where complex per‑node resource allocations must be respected without sacrificing overall cost efficiency.