Computing NP-Hardness of p-Presentation Distances in Merge Trees and Persistence Modules

Core Concepts

The authors demonstrate the NP-hardness of computing p-presentation distances for merge trees and t-parameter persistence modules, providing a novel strategy applicable to various settings in topological data analysis.

Abstract

The content discusses the introduction of p-presentation distances for merge trees and multiparameter persistence modules, highlighting their sensitivity compared to interleaving distances. The authors prove the NP-hardness of computing these distances for various values of p, showcasing a novel approach that can be extended to other distance metrics based on sums or p-norms. Merge trees and persistence modules are fundamental concepts in topological data analysis, with applications in data visualization and homology studies. The article presents a detailed proof strategy involving algebraic and combinatorial objects, emphasizing the complexity of computing these distances. The study also acknowledges contributions from related research areas and provides an outline for further exploration.

Stats

It is well-known that computing the interleaving distance is NP-hard in both cases.
Computing the p-presentation distance is NP-hard for all p ∈ [1, ∞) for both merge trees and t-parameter persistence modules for any t ≥ 2.
For single-parameter persistence, the interleaving distance is equal to the bottleneck distance.
Recently, there has been an increasing interest in studying the homology of filtrations dependent on more than a single variable.
These distances are called p-presentation distances, and it was shown that these distances are all universal in their respective settings.
The problems of computing the p-presentation distance for merge trees and t-parameter persistence modules for any t ≥ 2 are NP-hard for all p ∈ [1, ∞).
Our two proofs are summarized in Theorem 1.1.

Quotes

Key Insights Distilled From

Computing $p$-presentation distances is hard

by Håva... at arxiv.org 03-13-2024

https://arxiv.org/pdf/2403.07200.pdf

Deeper Inquiries

How can the concept of p-presentation distances be applied to other fields beyond topological data analysis

The concept of p-presentation distances, as explored in the context of topological data analysis, can be applied to various other fields beyond TDA. One potential application is in machine learning and pattern recognition, where these distances can be used to compare and analyze complex datasets. By incorporating p-presentation distances into algorithms for clustering or classification tasks, researchers can potentially improve the accuracy and efficiency of these processes. Additionally, in computational biology, these distances could aid in comparing biological sequences or structures, leading to advancements in genomics and bioinformatics research.

What potential implications could arise from efficiently computing these distances

Efficiently computing p-presentation distances has several potential implications across different domains. In TDA specifically, it could enhance the analysis of high-dimensional data sets by providing more nuanced insights into the shape and structure of the data. This could lead to improved visualization techniques and better understanding of complex relationships within the data. Moreover, in applications such as image processing or signal analysis, efficient computation of these distances could enable faster processing times and more accurate results.

How do these findings contribute to advancing computational complexity theory

These findings contribute significantly to advancing computational complexity theory by establishing NP-hardness results for computing p-presentation distances in merge trees and multiparameter persistence modules. By demonstrating that these computations are computationally challenging even for specific values of p within a defined range (from 1 to infinity), this research adds valuable insights into the complexity landscape of distance calculations based on sums or norms. The novel strategies employed in proving NP-hardness open up avenues for further exploration into hardness proofs for other distance metrics used across various disciplines involving maxima and sums.