Distance problems within Helly graphs and $k$-Helly graphs

TL;DR

This paper investigates distance problems in Helly and $k$-Helly graphs, providing efficient algorithms for computing central vertices, medians, and radius, with extensions to approximate solutions under relaxed Helly conditions.

Contribution

It introduces new algorithms for key distance computations in Helly graphs, extending to $k$-Helly graphs and approximate solutions with improved time complexity.

Findings

Central vertices and medians can be computed in $ ilde{O}(m\sqrt{n})$ time.

Radius computation in $k$-Helly graphs can be done in $ ilde{O}(m\sqrt{kn})$ time.

Approximate radius algorithms with constant additive error are achievable under relaxed Helly conditions.

Abstract

The ball hypergraph of a graph $G$ is the family of balls of all possible centers and radii in $G$. It has Helly number at most $k$ if every subfamily of $k$-wise intersecting balls has a nonempty common intersection. A graph is $k$-Helly (or Helly, if $k=2$) if its ball hypergraph has Helly number at most $k$. We prove that a central vertex and all the medians in an $n$-vertex $m$-edge Helly graph can be computed w.h.p. in $\tilde{\cal O}(m\sqrt{n})$ time. Both results extend to a broader setting where we define a non-negative cost function over the vertex-set. For any fixed $k$, we also present an $\tilde{\cal O}(m\sqrt{kn})$-time randomized algorithm for radius computation within $k$-Helly graphs. If we relax the definition of Helly number (for what is sometimes called an "almost Helly-type" property in the literature), then our approach leads to an approximation algorithm for computing the radius with an additive one-sided error of at most some constant.