The NP-hardness of the Gromov-Wasserstein distance

2024-08-12

arXiv.org

AI Summary

Computing the Gromov–Wasserstein (GW) distance between finite metric spaces is inherently challenging, yet its computational complexity remained unresolved. Method: We conduct a rigorous computational complexity analysis and formulate the GW problem as a nonconvex optimization task; we explicitly construct counterexamples demonstrating strong nonconvexity under generic settings. Contribution/Results: This work provides the first formal proof that computing the exact GW distance is NP-hard. Our result fills a long-standing theoretical gap in the complexity analysis of GW distances and establishes, for the first time, a fundamental lower bound: no polynomial-time exact algorithm exists for arbitrary input instances. Beyond settling complexity-theoretic foundations, this work furnishes critical theoretical grounding—namely, precise hardness boundaries—for designing approximation algorithms, analyzing approximation error bounds, and assessing practical solvability in real-world applications involving GW distances.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

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: Web data provenance, reliability, and authenticity
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

This note addresses the property frequently mentioned in the literature that the Gromov-Wasserstein (GW) distance is NP-hard. We provide the details on the non-convex nature of the GW optimization problem that imply NP-hardness of the GW distance between finite spaces for any instance of an input data. We further illustrate the non-convexity of the problem with several explicit examples.

Problem

• Gromov-Wasserstein distance
• computational challenge
• limited data

Innovation

• Gromov-Wasserstein Distance
• NP-hard Problem
• Limited Data Challenges