Given a set of areas, draw a diagram of regions with those areas minimizing the total perimeter. This appears to be a problem for which no known algorithm is guaranteed to solve. In three dimensions with only two regions, the solution is the standard double bubble.
Assuming a solution, form a graph of adjacent regions. (This graph could be complicated with toroidal regions.) Given just this graph, how difficult is it to reconstruct the shape and borders of each region? I suspect this becomes an easier local minimization problem.
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