The Complexity of Metric Realization

It is shown that the problem of realizing a metric by a graph or network with minimum total edge-length is, depending on the version, NP-hard or NP-complete.In particular, Discrete Metric Realization (DMR) is NP-complete “in the strong sense,” where DMR is defined as follows:INSTANCE. An n-by-n integer-entry distance matrix $D = ( d_{i,j} )$ and a positive integer k;QUESTION. Is there a graph $G = \langle V,E \rangle $ with distinguished vertices $v _1 ,v _2 , \cdots ,v _n $ that realize D, i.e., the number of edges in a shortest path between $v _i $ and $v _j $ is exactly $d_{i,j} $ for each $1\leqq i\leqq j\leqq n$, and such that the total number of edges of G is at most k?