Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimal Unimodal Decomposition is NP-Hard on Graphs

Mishal Assif P K, Yuliy Baryshnikov

TL;DR

The paper establishes that computing minimal unimodal decompositions, as captured by $ucat^p(f)$, is NP-hard on graphs with cycles and extends these hardness results to planar graphs and higher-dimensional simplicial complexes. It shows tractability on trees via a greedy $O(|V|^2)$ algorithm, but proves NP-hardness by reductions from Graph $k$-coloring and Vertex Cover for general graphs, including planarity-preserving variants and approximation boundaries (hard to beat a factor of $\sqrt{2}$). In higher dimensions, unimodality can be undecidable for $d\ge4$, while NP-hardness persists for $d=2,3$ through simplicial embeddings of planar graphs. The results delineate a transition from efficient tree-based computation to intractability in more complex topologies, guiding future work on approximations and fixed-parameter strategies.

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.

Minimal Unimodal Decomposition is NP-Hard on Graphs

TL;DR

The paper establishes that computing minimal unimodal decompositions, as captured by , is NP-hard on graphs with cycles and extends these hardness results to planar graphs and higher-dimensional simplicial complexes. It shows tractability on trees via a greedy algorithm, but proves NP-hardness by reductions from Graph -coloring and Vertex Cover for general graphs, including planarity-preserving variants and approximation boundaries (hard to beat a factor of ). In higher dimensions, unimodality can be undecidable for , while NP-hardness persists for through simplicial embeddings of planar graphs. The results delineate a transition from efficient tree-based computation to intractability in more complex topologies, guiding future work on approximations and fixed-parameter strategies.

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 is the smallest number of unimodal functions that sum up to . 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 , we establish the NP-hardness of finding a unimodal decomposition consisting of 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.

Paper Structure

This paper contains 10 sections, 17 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Categories and Coordinate-Free Generalizations
  3. Unimodality
  4. Functions on Graphs
  5. Outline of Results
  6. Minimal Unimodal Decomposition on Trees
  7. Minimal Unimodal Decomposition on General Graphs
  8. Strongly Unimodal Decomposition on General Graphs
  9. Minimal Unimodal Decomposition in Higher Dimensions
  10. Conclusion and Future Work

Key Result

Theorem 2.1

Let $G$ be a tree and $f : G \rightarrow [0,\infty)$ be a edge linear unimodal function on $G$. If $v_{max}$ is a vertex at which $f$ is maximized and the tree is rooted at $v_{max}$, then $f$ is non-increasing away from the root to its leaves.

Figures (1)

  • Figure 1: A graph $G$ (left) and the function $\tilde{f}$ on the augmented graph $\tilde{G}$, as defined in the proof of Theorem \ref{['th:main2']} (right).

Theorems & Definitions (34)

  • Definition 1.1: Unimodal Function
  • Definition 1.2: Unimodal Category
  • Definition 1.3: Strong Unimodal Category
  • Remark 1.4
  • Theorem 2.1
  • Theorem 2.2: Corollary 6.1 Bar20
  • Theorem 3.1
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['th:main']} when $k = 1$
  • proof : Proof of Theorem \ref{['th:main']} when $k \geq 3$
  • ...and 24 more