Label coloring to maximize number of "balanced" triangles (NP-hardness)
Reduce from 2-Lin(2):

Start with a star that has one edge per variable, and let those initial-labels all be 0.

For each equation ​ ​ ​ xi + xj ​ = ​ b ​ , ​ ​ ​ create an edge from the outer vertex of xi's edge to the outer vertex of xj's edge, and let that edge's initial-weight be (-1)b.

That will produce the output weighted-graph. ​ Just let k be the number of variables, so the cardinality constraint doesn't matter.

Triangles whose weights are all forced are forced to be good or bad, so their existence affects approximation quality but not which g's are optimal. ​ Other triangles use at least one variable-edge, and so exactly 2 variable-edges, since being a variable-edge is equivalent to being incident with the central vertex. ​ Thus no unintended triangles affect which g's are optimal, so those g's are exactly the g's for which composing with log-1 yields an optimal solution to the input 2-Lin(2) instance.



Your problem is obviously in W[P]O, and I have no other clue regarding its parameterized complexity.