Computability of width of submodular partition functions

The notion of submodular partition functions generalizes many of well-known tree decompositions of graphs. For fixed kk, there are polynomial-time algorithms to determine whether a graph has tree-width, branch-width, etc. at most kk. Contrary to these results, we show that there is no sub-exponential algorithm for determining whether the width of a given submodular partition function is at most two.