I'm wondering if anyone knows any efficient algorithms for finding the optimal placement of weights around a circle to minimize the center of mass. The mathematical formulation is as follows:
$$\min_{\sigma\in S_n} \left|\sum_{k=0}^{n-1} m_{\sigma(k)} e^{\frac{2\pi ik}{n}}\right|^2,$$ where $S_n$ is the set of permutations on $n$ elements and weights $m_0,m_1,...,m_{n-1}\in\mathbb{R}$ are given. The context is an engineering problem, so if there is a heuristic or stochastic method that gives a near optimal solution that would be fine.
I've considered a brute force check, but the number of permutations one would need to check to be comprehensive is $\frac{(n-1)!}{2}$ (considering the fact that reflections and rotations give the same weight distribution).
This is a "2-dimensional" variant of the partition problem (in a nutshell: split a set of numbers as evenly as possible). As such, on the one hand the problem seems NP-hard; on the other hand it seems that the approximation schemes for the partition problem should not be too hard to adapt.