The Homogeneous Broadcast Problem in Narrow and Wide Strips II: Lower Bounds

Let P be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $$s\in P$$sýP 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 $${\mathsf {W[1]}}$$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 $$f(k)n^{o(\sqrt{k})}$$f(k)no(k), unless ETH fails. The construction can also be used to show an $$f(w)n^{\varOmega (w)}$$f(w)nΩ(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).