Degree Bounded Matroids and Submodular Flows

We consider two related problems, the Minimum Bounded Degree Matroid Basis problem and the Minimum Bounded Degree Submodular Flow problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree problem: we are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e) ≤ g(e) for each hyperedge e. The task is to find a minimum cost basis which contains at least f(e) and at most g(e) elements from each hyperedge e. In the second problem we have a submodular flow problem, a lower bound f(v) and an upper bound g(v) for each node v, and the task is to find a minimum cost 0-1 submodular flow with the additional constraint that the sum of the incoming and outgoing flow at each node v is between f(v) and g(v). Both of these problems are NP-hard (even the feasibility problems are NP-complete), but we show that they can be approximated in the following sense. Let opt be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B of cost no more than opt such that f(e) − 2Δ + 1 ≤ |B ∩ e| ≤ g(e) + 2Δ − 1 for every hyperedge e, where Δ is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to Δ − 1. For the second problem we can find a 0-1 submodular flow of cost at most opt where the sum of the incoming and outgoing flow at each node v is between f(v) − 1 and g(v) + 1. These results can be applied to obtain approximation algorithms for different combinatorial optimization problems with degree constraints, including the Minimum Crossing Spanning Tree problem, the Minimum Bounded Degree Spanning Tree Union problem, the Minimum Bounded Degree Directed Cut Cover problem, and the Minimum Bounded Degree Graph Orientation problem.