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Article Dans Une Revue Theoretical Computer Science Année : 2023

Distance problems within Helly graphs and k-Helly graphs

Résumé

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 Õ(m √ n) 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 Õ(m √ 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.
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Dates et versions

hal-03942267 , version 1 (17-01-2023)

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Guillaume Ducoffe. Distance problems within Helly graphs and k-Helly graphs. Theoretical Computer Science, 2023, 946, pp.113690. ⟨10.1016/j.tcs.2023.113690⟩. ⟨hal-03942267⟩
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