New Bounds for Approximating Extremal Distances in Undirected Graphs

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−ε)-approximation of the diameter or a (5/3 − ε)-approximation of all the eccentricities in O(m 2−δ) time for any ε, δ > 0, even allowing for a constant additive term in the approximation. Second, we present an algorithmic scheme that gives a (2 − 1/2 k)-approximation of the diameter and the radius and a (3 − 4/(2 k + 1))-approximation of all eccentricities in O(mn 1 k+1) 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.
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Massimo Cairo, Roberto Grossi, Romeo Rizzi. New Bounds for Approximating Extremal Distances in Undirected Graphs. SIAM: ACM-SIAM Symposium on Discrete Algorithms (SODA'16), Jan 2016, Arlington, United States. pp.363 - 376, ⟨10.1137/1.9781611974331.ch27⟩. ⟨hal-01526665⟩

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