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 for radius, diameter and all eccentricities is a barrier against SETH-based lower bounds for these problems. We indeed prove that for graph classes with certificates of bounded size, 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. In particular, our formalization enables a novel primal-dual analysis of a classical approach for diameter computation. Based on our novel insights for these problems, we introduce several new algorithmic techniques related to eccentricity computation and propose 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 in the case where the input graph is a power-law random graph, has low doubling dimension, has low hyperbolicity, is chordal, satisfies some Helly-type property, or has bounded asteroidal number.

翻译：在细粒度复杂性背景下，我们研究能够实现更快多项式时间算法的证书概念。我们特别针对半径（最小偏心距）、直径（最大偏心距）及所有偏心距的计算问题展开研究，这些问题的二次时间下界在合理猜想下已被证实。针对每种情况，我们引入一种证书概念，将其定义为一组特定节点，当该集合具有次线性规模时，可以从这些节点在次二次时间内推导出所有偏心距的适当边界。半径、直径和所有偏心距存在小型证书这一事实，构成了针对这些问题基于SETH下界的理论障碍。我们严格证明：对于具有有界规模证书的图类，分别存在计算半径、直径及所有偏心距的随机化次二次时间算法。此外，这些证书概念与通过单源全距离查询探测图的算法紧密相关，并能解释文献中实用半径与直径算法的效率来源。特别地，我们的形式化框架为经典直径计算方法提供了新颖的原对偶分析。基于对这些问题的新见解，我们引入了若干与偏心距计算相关的新算法技术，并提出了针对半径、直径及所有偏心距的算法，这些算法在特定图参数方面具有理论保证。我们通过对各类实际图数据的实验验证了这些参数在实践中通常取值较低。最后，我们对输入图为幂律随机图、具有低倍维数、具有低双曲性、是弦图、满足某些Hel利型性质或具有有界星状数的情况，获得了更精细的结果。