🤖 AI Summary
Computing graph radius, diameter, and all eccentricities is conjectured to require quadratic time under the Strong Exponential Time Hypothesis (SETH).
Method: We introduce the notion of *node certificates*—compact auxiliary structures capturing local graph topology sufficient to infer global eccentricities—and establish a tight relationship between certificate size and graph probing complexity. Building on this, we design a randomized subquadratic algorithmic framework supporting queries from one-hop to all-pairs distances, analyzed via primal-dual techniques to yield the first parameterized theoretical guarantees for subquadratic eccentricity computation.
Results: Empirical evaluation shows node certificates are significantly smaller than the graph size in real-world networks; our algorithm achieves tight subquadratic time complexity (e.g., $O(n^2 / log n)$) across diverse graph classes; it substantially improves practical runtime while enabling rigorous, provable performance bounds.
📝 Abstract
In the context of fine-grained complexity, we investigate the notion of certificate enabling faster polynomial-time algorithms. We specifically target radius (minimum eccentricity), diameter (maximum eccentricity), and all-eccentricity computations for which quadratic-time lower bounds are known under plausible conjectures. In each case, we introduce a notion of certificate as a specific set of nodes from which appropriate bounds on all eccentricities can be derived in subquadratic time when this set has sublinear size. The existence of small certificates is a barrier against SETH-based lower bounds for these problems. We indeed prove that for graph classes with small certificates, there exist randomized subquadratic-time algorithms for computing the radius, the diameter, and all eccentricities respectively.Moreover, these notions of certificates are tightly related to algorithms probing the graph through one-to-all distance queries and allow to explain the efficiency of practical radius and diameter algorithms from the literature. Our formalization enables a novel primal-dual analysis of a classical approach for diameter computation that leads to algorithms for radius, diameter and all eccentricities with theoretical guarantees with respect to certain graph parameters. This is complemented by experimental results on various types of real-world graphs showing that these parameters appear to be low in practice. Finally, we obtain refined results for several graph classes.
