An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection

Given an undirected graph G = (V, E) with |V| = n and |E| = m, nonnegative integers c_e and d_e for each edge e ∈ E, and a bound D, the constrained minimum spanning tree problem (CST) is to find a spanning tree T = (V, E_T) such that Σ_{e ∈ E T} d _e ≤ D and Σ_{e∈E T} c_e is minimized. We present an efficient polynomial time approximation scheme (EPTAS) for this problem. Specifically, for every ε > 0 we present a (1 + ε)-approximation algorithm with time complexity O((1/ε) ^O(1/ε) n⁴). Our method is based on Lagrangian relaxation and matroid intersection.