In order to formulate the given question as a set cover problem, let's assume that $T^{*}$ to be the set of all triangles in the graph.

$T^{*}=\{(e_{i}, e_{j}, e_{k})\in E^{3}: e_{i}, e_{j}, e_{k}$ form a triangle$\}$

For each $e$, let $T_{e}$ denote the set of all triangles containing $e$. We define the weight of each each set $T_{e}$ to be the weight of its corresponding edge $e$ ($w(T_{e})=w(e)$). In result of these changes, the problem of interest would become finding the minimum total weight sub-collection of $T_{e}$'s that cover $T^{*}$. And this is exactly the problem of weighted set cover.

In sum, we could formulate the problem of *finding the minimum weight set of edges that destroy triangles* as the problem of *finding the minimum total weight subcollection of $T_{e}$'s that cover $T^{*}$ *.

real-valued edge-weights, as your notation "$w(e)\geq0$" seems to suggest? (2) do you really mean "set cover problem" in thestandardsense (i.e. set of edges of a hypergraph whose union equals the ground-set)? The reason for my asking (2) is that, for reasons I can't go into in a comment, your problem feelsmuchmore like the (somewhat dual) problem ofminimum vertex covers(synonym:transversal; synonym:hitting set). I recall that if combinatorialists say 'X cover' they mean 'cover consisting of Xs'. $\endgroup$ – Peter Heinig Apr 15 '18 at 18:10