Recognizing Distance-Count Matrices is Difficult

2025-08-26

AI Summary

This paper investigates the realizability problem for Distance Counting Matrices (DCMs) of graphs—integer matrices whose $(i,j)$-entry counts the number of node pairs at graph distance $d$. The authors establish, for the first time, that deciding whether a given integer matrix is the DCM of some undirected simple graph is strongly NP-complete. This result is rigorously proven via a polynomial-time reduction from the 3-Partition problem. The strong NP-completeness implies that no polynomial-time algorithm exists unless P = NP, thereby establishing a fundamental theoretical barrier for axiomatizing geometric centrality measures—particularly in constructing counterexamples. Moreover, it indicates that synthesizing graphs with prescribed distance distributions necessitates intelligent approaches beyond brute-force enumeration, such as heuristic search or constrained optimization.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Axiomatization of centrality measures often involves proving that something cannot hold by providing a counterexample (i.e., a graph for which that specific centrality index fails to have a given property). In the context of geometric centralities, building such counterexamples requires constructing a graph with specific distance counts between nodes, as expressed by its distance-count matrix. We prove that deciding whether a matrix is the distance-count matrix of a graph is strongly NP-complete. This negative result implies that a brute-force approach to building this kind of counterexample is out of question, and cleverer approaches are required.

Problem

• Determining if a matrix represents graph node distances
• Proving constructing counterexamples for centrality is difficult
• Establishing the NP-completeness of distance-count matrix recognition

Innovation

• Proving NP-completeness of matrix recognition
• Analyzing distance-count matrices for graphs
• Rejecting brute-force counterexample construction methods