Minimal Unimodal Decomposition is NP-Hard on Graphs

2025-10-07

AI Summary

This paper studies the minimum unimodal decomposition problem for edge-wise linear functions on graphs: representing such a function as the sum of the fewest unimodal functions—i.e., functions whose superlevel sets are contractible. Motivated by structured modeling of density functions in topological statistics, the authors establish, for the first time, that computing the minimum unimodal decomposition is NP-hard even on simple graphs. Moreover, for any fixed $k geq 2$, deciding whether a decomposition into $k$ unimodal functions exists is also NP-hard. These hardness results extend to planar graphs, admit no polynomial-time constant-factor approximation (i.e., are APX-hard), and generalize to analogous decomposition problems on higher-dimensional simplicial complexes. Technically, the proofs employ intricate reductions that encode known NP-hard problems into the topological constraint of contractibility of superlevel sets, carefully leveraging interplay between graph structure and topological collapsibility. This work systematically establishes computational lower bounds for unimodal decomposition in topological statistics.

Technology Category

Reasoning under Uncertainty: Graphical ModelsKnowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsWeb Mining and Content Analysis: Bridging structured and unstructured dataEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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.

Problem

• Finding minimal unimodal decomposition is NP-hard on graphs
• The problem extends to planar graphs and higher dimensions
• Inapproximability results hold for k-component decompositions

Innovation

• Proves minimal unimodal decomposition NP-hard
• Establishes NP-hardness for k-component decompositions
• Extends NP-hardness to planar graphs and variants