Distance problems within Helly graphs and k-Helly graphs

Abstract

The ball hypergraph of a graph G is the family of balls of all possible centers and radii in G. Balls in a subfamily are k-wise intersecting if the intersection of any k balls in the subfamily is always nonempty. The Helly number of a ball hypergraph is the least integer k greater than one such that 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 O˜(mn) time. Both results extend to a broader setting where we assign a nonnegative cost to the vertices. For any fixed k, we also present an O˜(mkn)-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.