I was imagining a simpler model, where different areas of the beard are either present or not; length doesn't matter. If you're growing one part out, you might

I was imagining a simpler model, where different areas of the beard are either present or not; length doesn't matter. If you're growing one part out, you might as well grow all of it out. Therefore, in this model, you only "grow" to the full beard vertex. I'm not sure that your reduction from TSP works. Given two nodes A and B, I understand that you're setting their distance on the associated facial areas to match the distance in TSP, but what if they're forced to be even more distant on the area corresponding to some other node C?