New Bounds for Approximating Extremal Distances in Undirected Graphs

This Paper Appears in Cover Image Title Information Published: 2016 eISBN: 978-1-61197-433-1 Book Code: PRDA16 Series: Proceedings Pages: 14 Abstract We provide new bounds for the approximation of extremal distances (the diameter, the radius, and the eccentricities of all nodes) of an undirected graph with n nodes and m edges. First, we show under the Strong Exponential Time Hypothesis (SETH) of Impagliazzo, Paturi and Zane [JCSS01] that it is impossible to get a (3/2 – ∊)-approximation of the diameter or a (5/3 – ∊)-approximation of all the eccentricities in O(m2–δ) time for any ∊, δ > 0, even allowing for a constant additive term in the approximation. Second, we present an algorithmic scheme that gives a (2 – 1/2k)-approximation of the diameter and the radius and a (3 – 4/(2k + 1))-approximation of all eccentricities in expected time for any k ≥ 0. For k ≥ 2, this gives a family of previously unknown bounds, and approaches near-linear running time as k grows. Third, we observe a connection between the approximation of the diameter and the h-dominating sets, which are subsets of nodes at distance ≤ h from every other node. We give bounds for the size of these sets, related with the diameter.