Computability of Width of Submodular Partition Functions

摘要

The notion of submodular partition functions generalizes many of well-known tree decompositions of graphs. For fixed k, there are polynomial-time algorithms to determine whether a graph has tree-width, branch-width, etc. at most k. 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. In addition, we also develop another dual notion for submodular partition functions which is analogous to loose tangles for connectivity functions.