A matter of degree: improved approximation algorithms for degree-bounded minimum spanning trees

In this paper, we present a new bicriteria approximation algorithm for the degree-bounded minimum spanning tree problem. In this problem, we are given an undirected graph, a nonnegative cost function on the edges, and a positive integer B*, and the goal is to find a minimum-cost spanning tree T with maximum degree at most B*. In an n-node graph, our algorithm finds a spanning tree with maximum degree O(B*+logn) and cost O(optB*), where optB* is the minimum cost of any spanning tree whose maximum degree is at most B*. Our algorithm uses ideas from Lagrangean duality. We show how a set of optimum Lagrangean multipliers yields bounds on both the degree and the cost of the computed solution.