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

TL;DR

The paper develops a formal certificate framework for radius, diameter, and all eccentricities in graphs, tying the existence of small certificates to subquadratic-time algorithms via one-to-all distance queries. It defines radius/dia/all-ecc certificates, extended diameter certificates, and antipodes, and connects certificate size to set-cover complexities. It then shows subquadratic algorithms for graphs with small certificates, provides practical radius/diameter/all-eccentricities algorithms, and demonstrates that real-world graphs often admit very small certificates. The study further analyzes several graph classes (power-law, doubling-dimension, hyperbolic, chordal, Helly, asteroidal-number-bounded) where certificates yield strong algorithmic benefits, thereby clarifying when the quadratic barrier can be breached in P for these problems.

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.

Figures (1)

• Figure 1: An example of bow-tie shaped graph $BT_{p,q}$ (for $p\ge 2$ and $q\ge 6$) with radius certificate $R$, diameter certificate $D$ and all eccentricity certificate ($R,U$). Its diameter is $4q-2$ (eccentricity of blue nodes). Its radius is $2q+1$ (eccentricity of green nodes). Plain lines correspond to edges while a dashed line with label $\ell$ corresponds to a path of length $\ell$. All unlabeled dashed lines from $a$ and $b$ have length 2, while all unlabeled dashed lines from blue nodes have length $q-2$ (similarly to given labels on upper lines).

Theorems & Definitions (58)

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