I was amused by the work of Scott Aaronson on soap bubbles and Steiner trees. Given $n$ points $p_1$, $p_2$, ..., $p_n$ in $\mathbb{R}^2$, the [Steiner tree][1] through these points is the connected planar graph of shortest length containing these points. Computing the topology of the Steiner tree, given the coordinates of the points, is NP-hard.

On the other hand, if two parallel glass plates are separated by rods in positions $p_1$, $p_2$, ..., $p_n$ and dipped into a bucket of soapy water, the shape of the resulting soap film is a local minimum for this problem. Which lead to the question: How close is the local minimum, formed by whatever complicated PDE governs soap film, to the NP-hard global minimum?

Aaronson decided to try the experiment, and reported on his results in Section 3 of "[NP-complete problems and physical reality][2]". The answer was not very close: The stable shape of the soap film was often not even a tree!

[![enter image description here][3]][3]


  [1]: https://en.wikipedia.org/wiki/Steiner_tree_problem
  [2]: https://arxiv.org/abs/quant-ph/0502072
  [3]: https://i.sstatic.net/EPUoA.jpg