Hardness of Dynamic Core and Truss Decompositions

2025-09-16

AI Summary

This paper investigates the computational complexity of dynamic *k*-core and *k*-truss decomposition in graphs. Leveraging the OMv (Online Matrix-Vector multiplication) and SETH (Strong Exponential Time Hypothesis) conjectures, it establishes the first rigorous conditional lower bounds for their dynamic maintenance: under edge updates, no algorithm can compute exact or (2−ε)-approximate core numbers asymptotically faster than static reconstruction—i.e., in *o*(*m*) time—where *m* is the number of edges. These hardness results extend to directed graphs and *k*-truss variants. Notably, efficiency remains impossible even under bounded update sequences or restricted change regimes. The sole exception is the 2-core, which admits *O*(polylog *n*)-time dynamic updates. By reducing dynamic *k*-core/*k*-truss maintenance to fundamental problems—including matrix multiplication and SAT—the work systematically uncovers the intrinsic hardness of these decompositions and establishes a theoretical ceiling for future algorithmic development.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Computational social science
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The k-core of a graph is its maximal subgraph with minimum degree at least k, and the core value of a vertex u is the largest k for which u is contained in the k-core of the graph. Among cohesive subgraphs, k-core and its variants have received a lot of attention recently, particularly on dynamic graphs, as reported by Hanauer, Henzinger, and Schulz in their recent survey on dynamic graph algorithms. We answer questions on k-core stated in the survey, proving that there is no efficient dynamic algorithm for k-core or to find (2 - ε)-approximations for the core values, unless we can improve decade-long state-of-the-art algorithms in many areas including matrix multiplication and satisfiability, based on the established OMv and SETH conjectures. Some of our results show that there is no dynamic algorithm for k-core asymptotically faster than the trivial ones. This explains why most recent research papers in this area focus not on a generic efficient dynamic algorithm, but on finding a bounded algorithm, which is fast when few core values change per update. However, we also prove that such bounded algorithms do not exist, based on the OMv conjecture. We present lower bounds also for a directed version of the problem, and for the edge variant of the problem, known as k-truss. On the positive side, we present a polylogarithmic dynamic algorithm for 2-core.

Problem

• Proving no efficient dynamic k-core decomposition algorithms exist
• Establishing lower bounds for k-core and k-truss approximations
• Demonstrating impossibility of bounded dynamic core algorithms

Innovation

• Proves no efficient dynamic k-core algorithm exists
• Shows bounded algorithms impossible under OMv
• Provides polylogarithmic algorithm for 2-core