The Homogeneous Broadcast Problem in Narrow and Wide Strips

Let $P$ be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $s\in P$ 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), with the restriction that all points lie inside a strip of width $w$. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width $w$.Comment: 50 pages, WADS 2017 submissio

The homogeneous broadcast problem in narrow and wide strips

Let $P$ be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $s\in P$ 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), with the restriction that all points lie inside a strip of width $w$. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width $w$

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∈ 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). </p

The Homogeneous Broadcast Problem in Narrow and Wide Strips I: Algorithms

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), with the restriction that all points lie inside a strip of width w. We describe several algorithms for both the regular and the hop-bounded versions, and show that both problems are solvable in polynomial time in strips of small constant width. These results complement the hardness results in a companion paper (de Berg et al. in Algorithmica, 2017)

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∈ 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)