What Would Edmonds Do? Augmenting Paths and Witnesses for Degree-Bounded MSTs

Given a graph and degree upper bounds on vertices, the BDMST problem requires us to find the minimum cost spanning tree respecting the given degree bounds.Konemann and Ravi [10,11] give bicriteria approximation algorithms for the problem using local search techniques of Fischer [5]. For a graph with a cost C, degree B spanning tree, and parameters b, w> 1, their algorithm produces a tree whose cost is at most wC and whose degree is at most $\frac{w}{w-1}bB + \log_b n.$ We give a polynomial-time algorithm that finds a tree of optimal cost and with maximum degree at most bB + 2(b+1)logbn. We also give a quasi-polynomial algorithm which produces a tree of optimal cost C and maximum degree bounded by B + O(log n/loglog n). Our algorithms work when there are upper as well as lower bounds on the degrees of the vertices.