Recognizing Distance-Count Matrices is Difficult

TL;DR

Determining whether a given matrix can represent the distance counts of a graph is computationally hard, specifically NP-complete, complicating the construction of counterexamples in geometric centrality measures.

Contribution

We prove that recognizing distance-count matrices of graphs is strongly NP-complete, highlighting the computational difficulty in constructing counterexamples for geometric centralities.

Findings

Recognition problem is strongly NP-complete

Implication: brute-force counterexample construction is infeasible

Necessity for more sophisticated methods in geometric centrality analysis

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.