Certificates in P and Subquadratic-Time Computation of Radius, Diameter, and all Eccentricities in Graphs
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.