The homogeneous broadcast problem in narrow and wide strips II: lower bounds

Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak (Corresponding author)

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Abstract

Let P be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let s∈ P be a given source node. Each node p can transmit information to all other nodes within unit distance, provided p is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source s can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem—in the latter s must be able to reach every node within a specified number of hops—where we also consider how the complexity depends on the width w of the strip. We prove the following two lower bounds. First, we show that the regular version of the problem is W[1] -complete when parameterized by the solution size k. More precisely, we show that the problem does not admit an algorithm with running time f(k)no(k), unless ETH fails. The construction can also be used to show an f(w) n Ω ( w ) lower bound when we parameterize by the strip width w. Second, we prove that the hop-bounded version of the problem is NP-hard in strips of width 40. These results complement the algorithmic results in a companion paper (de Berg et al. in Algorithmica, submitted).

Original languageEnglish
Pages (from-to)2963-2990
Number of pages28
JournalAlgorithmica
Volume81
Issue number7
Early online date19 Mar 2019
DOIs
Publication statusPublished - Jul 2019

Keywords

  • Broadcast
  • Dominating set
  • Range assignment
  • Unit disk graph

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