Hardness of covering all the triangles

Vertex Cover and Edge Cover are two classical examples that are often used to show the contrast of problem solvability. While Vertex Cover is hard, Edge Cover can be solved in polynomial time. We claim that the former remains intractable even if the objects to be covered are triangles instead of edges. Therefore, one more combinatorial optimization problem, namely Covering Triangles, is added to the decades-old list of the problems in this research area.