Deciding if a DAG is Interesting is Hard

2025-03-17

AI Summary

This work studies the Interestingness Maximization problem (IP) for edge-disjoint paths in directed acyclic graphs (DAGs), along with its k-edge-constrained variant (k-IP, k ≥ 3), motivated by path selection in Mapper graphs from topological data analysis (TDA). The authors identify a flaw in prior NP-completeness proofs and provide the first rigorous polynomial-time reduction establishing that both IP and k-IP are NP-hard on DAGs. Their approach integrates computational complexity theory, graph algorithmics, and the Mapper framework from TDA. The main contributions are: (i) correction of a fundamental theoretical error in the literature; (ii) establishment of tight hardness lower bounds for these problems; and (iii) provision of essential theoretical foundations—including provable performance limits—for designing and analyzing heuristic algorithms targeting practical Mapper-based path extraction. (128 words)

Technology Category

• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The emph{interestingness score} of a directed path $Pi = e_1, e_2, e_3, dots, e_ell$ in an edge-weighted directed graph $G$ is defined as $ exttt{score}(Pi) := sum_{i=1}^ell w(e_i) cdot log{(i+1)}$, where $w(e_i)$ is the weight of the edge $e_i$. We consider two optimization problems that arise in the analysis of Mapper graphs, which is a powerful tool in topological data analysis. In the IP problem, the objective is to find a collection $mathcal{P}$ of edge-disjoint paths in $G$ with the maximum total interestingness score. %; that is, two raised to the power of the sum of the weights of the paths in $mathcal{P}$. For $k in mathbb{N}$, the $k$-IP problem is a variant of the IP problem with the extra constraint that each path in $mathcal{P}$ must have exactly $k$ edges. Kalyanaraman, Kamruzzaman, and Krishnamoorthy (Journal of Computational Geometry, 2019) claim that both IP and $k$-IP (for $k geq 3$) are NP-complete. We point out some inaccuracies in their proofs. Furthermore, we show that both problems are NP-hard in directed acyclic graphs.

Problem

• Assessing interestingness of paths in edge-weighted DAGs.
• Optimizing path collections for maximum interestingness score.
• Proving NP-hardness of IP and k-IP problems in DAGs.

Innovation

• Defines interestingness score for directed paths
• Analyzes Mapper graphs in topological data analysis
• Proves NP-hardness in directed acyclic graphs