Incremental Network Design with Minimum Spanning Trees

Given an edge-weighted graph $G=(V,E)$ and a set $E_0\subset E$, the incremental network design problem with minimum spanning trees asks for a sequence of edges $e_1^{\mathrm{\prime }},\dots ,e_T^{\mathrm{\prime }}\in E\setminus E_0$ minimizing ${\sum }_{t=1}^{T}w(X_t)$ where $w(X_t)$ is the weight of a minimum spanning tree $X_t$ for the subgraph $(V,E_0\cup \{e_1^{\mathrm{\prime }},\dots ,e_t^{\mathrm{\prime }}\})$ and $T=\lvert E\setminus E_0\rvert$. We prove that this problem can be solved by a greedy algorithm.