Polynomial-time computation of weak villainy

Determine whether the weak villainy Bw(Kp1,...,pk) of a complete multipartite graph can be computed in time polynomial in the binary encoding length of its part-size vector when the number of parts is part of the input.

Background

For a complete multipartite graph, the paper derives a finite ordered-potential formula for weak villainy and proves fixed-parameter tractability when the number of parts is fixed. The number of candidate potential shapes can nevertheless depend on the spread of the part sizes.

The unresolved computational question asks whether the dependence on the number of parts can also be removed, yielding an algorithm polynomial in the binary input length when both the part sizes and the number of parts vary.