Clearing Connections by Few Agents

Abstract

Abstract. We study the problem of clearing connections by agents placed at some vertices in a directed graph. The agents can move only along directed paths. The objective is to minimize the number of agents guaranteeing that any pair of vertices can be connected by a underlying undirected path that can be cleared by the agents. We provide several results on the hardness, approximability and parameterized complexity of the problem. In particular, we show it to be: NP-hard, 2-approximable in polynomial-time, and solvable exactly in O(αn 3 2 2α ) time, where α is the number of agents in the solution. In addition, we give a simple linear-time algorithm optimally solving the problem in digraphs whose underlying graphs are trees. Finally, we discuss a related problem, where the task is to clear with a minimum number of agents a subgraph of the underlying graph containing its spanning tree. We show that this problem also admits a 2-approximation in polynomial time.

Clearing directed subgraphs by mobile agents

“…On the other hand, we show that the ST problem (as well as some of its variants) is NP-complete, by a reduction from the Set Cover problem [24] (Section 3). Our result on NP-completeness of the ST problem implies NP-completeness of the Agent Clearing Tree problem studied in [33], where the complexity status of the latter has been posed as an open problem.…”

“…Since by setting F (v) = 1 for each vertex v of the input digraph, the STU problem becomes just the Agent Clearing Tree problem (ACT) studied in [33], we immediately obtain the following corollary resolving the open problem of the complexity status of ACT posed in [33].…”

“…Related work. The Snow Team problem is related to the problems of clearing connections by mobile agents placed at some vertices in a digraph, introduced by Levcopoulos et al in [33]. In particular, the ST problem is a generalized variant of the Agent Clearing Tree (ACT) problem where one wants to determine a placement of the minimum number of mobile agents in a digraph D such that agents, allowed to move only along directed walks, can simultaneously clear some subgraph of D whose underlying graph includes a spanning tree of the underlying graph of D. In [33], the authors provided a simple 2-approximation algorithm for solving the Agent Clearing Tree problem, leaving its complexity status open.…”