A push-relabel approximation algorithm for approximating the minimum-degree MST problem and its generalization to matroids

In the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to find a minimum spanning tree (MST) T , such that the maximum degree of T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST of degree at most 2∆opt(G)+ o(∆opt(G)), where ∆opt(G) denotes the degree of the optimal tree. This result improves on a previous result of Fischer [5] that finds an MST of degree at most b∆opt(G) + logb n, for any b > 1. The MDMST problem is a special case of the following problem: given a k-ary hypergraph G = (V,E) and weighted matroid M with E as its ground set, find a minimum-cost basis (MCB) T ofM such that the degree of T inG is as small as possible. Our algorithm immediately generalizes to this problem, finding an MCB of degree at most k2∆opt(G,M)+O(k √ k∆opt(G,M)). We use the push-relabel framework developed by Goldberg [8] for the maximum-flow problem. To our knowledge, this is the first use of the push-relabel technique in an approximation algorithm for an NP-hard problem. The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms for this problem by Könemann and Ravi [13, 14] and by Chaudhuri et al. [2] incur a near-logarithmic additive error in the degree. We give the first BDMST algorithm that approximates both the degree and the cost to within a constant factor of the optimal. These results generalize to the case of non-uniform degree bounds.