Minimal Unimodal Decomposition is NP-Hard on Graphs

A function on a topological space is called unimodal if all of its super-level sets are contractible. A minimal unimodal decomposition of a function $f$ is the smallest number of unimodal functions that sum up to $f$. The problem of decomposing a given density function into its minimal unimodal components is fundamental in topological statistics. We show that finding a minimal unimodal decomposition of an edge-linear function on a graph is NP-hard. Given any $k \geq 2$, we establish the NP-hardness of finding a unimodal decomposition consisting of $k$ unimodal functions. We also extend the NP-hardness result to related variants of the problem, including restriction to planar graphs, inapproximability results, and generalizations to higher dimensions.