Related papers: Certificates in P and Subquadratic-Time Computatio…
On sparse graphs, Roditty and Williams [2013] proved that no $O(n^{2-\varepsilon})$-time algorithm achieves an approximation factor smaller than $\frac{3}{2}$ for the diameter problem unless SETH fails. In this article, we solve an open…
The radius and diameter are fundamental graph parameters. They are defined as the minimum and maximum of the eccentricities in a graph, respectively, where the eccentricity of a vertex is the largest distance from the vertex to another…
The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter…
Among the most important graph parameters is the Diameter, the largest distance between any two vertices. There are no known very efficient algorithms for computing the Diameter exactly. Thus, much research has been devoted to how fast this…
We show that for any fixed integer $k \geq 0$, there exists an algorithm that computes the diameter and the eccentricies of all vertices of an input unweighted, undirected $n$-vertex graph of Euler genus at most $k$ in time \[…
Finding the diameter of a graph in general cannot be done in truly subquadratic assuming the Strong Exponential Time Hypothesis (SETH), even when the underlying graph is unweighted and sparse. When restricting to concrete classes of graphs…
We give the first truly subquadratic time algorithm, with $O^*(n^{2-1/18})$ running time, for computing the diameter of an $n$-vertex unit-disk graph, resolving a central open problem in the literature. Our result is obtained as an instance…
An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let $Ext_{\alpha}$ be the class of all connected graphs whose quotient graph obtained from modular…
We study $\textit{sparse singular value certificates}$ for random rectangular matrices. If $M$ is an $n \times d$ matrix with independent Gaussian entries, we give a new family of polynomial-time algorithms which can certify upper bounds on…
A graph algorithm is truly subquadratic if it runs in ${\cal O}(m^b)$ time on connected $m$-edge graphs, for some positive $b < 2$. Roditty and Vassilevska Williams (STOC'13) proved that under plausible complexity assumptions, there is no…
Local certification is a topic originating from distributed computing, where a prover tries to convince the vertices of a graph $G$ that $G$ satisfies some property $\mathcal{P}$. To convince the vertices, the prover gives a small piece of…
For a given computational problem, a certificate is a piece of data that one (the prover) attaches to the output with the aim of allowing efficient verification (by the verifier) that this output is correct. Here, we consider the minimal…
Median graphs form the class of graphs which is the most studied in metric graph theory. Recently, B\'en\'eteau et al. [2019] designed a linear-time algorithm computing both the $\Theta$-classes and the median set of median graphs. A…
Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs.…
This paper presents a distributed algorithm to simultaneously compute the diameter, radius and node eccentricity in all nodes of a synchronous network. Such topological information may be useful as input to configure other algorithms.…
We study the problem of efficiently certifying upper bounds on the independence number of $\ell$-uniform hypergraphs. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within…
The graph invariant twin-width was recently introduced by Bonnet, Kim, Thomass\'e, and Watrigan. Problems expressible in first-order logic, which includes many prominent NP-hard problems, are tractable on graphs of bounded twin-width if a…
The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each…
Kernel matrices, as well as weighted graphs represented by them, are ubiquitous objects in machine learning, statistics and other related fields. The main drawback of using kernel methods (learning and inference using kernel matrices) is…
We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a…