  Most of the literature on spanners focuses on building the graph from
scratch. This paper instead focuses on adding edges to improve an existing
graph. A major open problem in this field is: given a graph embedded in a
metric space, and a budget of $k$ edges, which $k$ edges do we add to produce a
minimum-dilation graph? The special case where $k=1$ has been studied in the
past, but no major breakthroughs have been made for $k > 1$. We provide the
first positive result, an $O(k)$-approximation algorithm that runs in $O(n^3
\log n)$ time.
