Reachability and Coverage Planning for Connected Agents: Extended Version

TL;DR

This paper investigates the complexity of multi-agent path planning with connectivity constraints, introducing sight-moveable graphs that enable efficient algorithms for reachability and coverage problems.

Contribution

It analyzes the complexity of connectivity-preserving path planning and introduces sight-moveable graphs as a new class with efficient solutions.

Findings

Complexity results for various graph classes

Introduction of sight-moveable graphs

Efficient algorithms for new graph class

Abstract

Motivated by the increasing appeal of robots in information-gathering missions, we study multi-agent path planning problems in which the agents must remain interconnected. We model an area by a topological graph specifying the movement and the connectivity constraints of the agents. We study the theoretical complexity of the reachability and the coverage problems of a fleet of connected agents on various classes of topological graphs. We establish the complexity of these problems on known classes, and introduce a new class called sight-moveable graphs which admit efficient algorithms.

Full text

1 Introduction

A number of use cases of planning rose in information-gathering missions from the development of unmanned autonomous vehicles (UAVs). For instance, in search and rescue missions, a fleet of drones can cover a lot of ground in a short amount of time and report any finding to a mission supervisor to narrow the search for the rescue team. Other examples are the analysis of terrain for smart farms and in hazardous locations. For this kind of missions, the information gathered is used for decision making at a supervising station. Thus, the drones need to be constantly in communication with the station to report the gathered information during the mission. The use of multiple UAVs to cover an area not only reduces the time required to complete the mission but can also enable reaching locations which would not be reachable with a single drone due to connection constraints. The original multi-agent path planning problem asks for a plan to reach a configuration of agents in a graph. However, an important problem for search and rescue missions or terrain analysis is the coverage of an area. We thus study both the coverage problem and the reachability problem under a connection constraint over the agents which requires them to be connected to the base either directly or via another agent, who can relay its data. We establish the computational complexity of the connected coverage in its general case and for a practical subclass introduced recently Tateo et al. (2018) in which the UAVs can communicate with others located within one step, called the neighbor-communicable topological graphs. We show that the coverage is PSPACE-complete in the general case, and remains so for neighbor-communicable topological graphs. Thus, restricting to neighbor-communicable graphs does not render the problem feasible, and the relatively high complexity unfortunately remains. Note that this is in line with the PSPACE-completeness of the reachability problem recently reported in Tateo et al. (2018).

Our main result in this paper is the definition of a class of topological graphs which is well adapted and realistic for UAV missions, and for which the coverage and reachability problems admit efficient solutions. Our subclass, called sight-moveable graphs, is defined assuming that the UAVs cannot communicate through obstacles and are restricted to line-of-sight communication. This class emerged from an ongoing case study for a drone assisted search and rescue project in which the authors take part. For this class, we prove that both the reachability and coverage problems are in LOGSPACE while the existence of a bounded execution is in NP. This drastically changes the status of this problem since by LOGSPACE $\subseteq$ NC (this is the class of problems solvable in polylogarithmic time in a parallel machine with a polynomial number of processors), one can build an efficient parallel algorithm Cook (1979). The NP upper bound is also useful since this means efficient SAT solvers can be used directly to compute bounded executions. We prove that our algorithms for the bounded variants are optimal by showing NP-hardness in each case.

In this work, we consider all agents to be anonymous in both cases (reachability and coverage). Furthermore, we consider the collisions to be handled by the agents themselves, hence are not considered along the results of this paper. We depicted a covering execution of a topological graph by 3 UAVs in Figure 1. In this example, the UAVs need to gather information at each node of the graph while staying connected to the base (red node) during the whole mission.

In Section 2, we present the typical notions used in MAPP and their extension for our case and the known results in connected planning. In Sections 3 to 6, we study the complexity of our problems from the general case to the most restrictive one. We describe the related works in Section 7. We conclude in Section 8.

2 Preliminaries

We first present the topological graphs and the subclasses we consider on which we study the complexity of our problems. Then, we give definitions of plans and executions, and formally define problems we consider.

In most applications of path planning, the space is discretized in order to generate a graph of movements on which algorithms are executed. For instance, regular grids which decompose the space in square, triangular or hexagonal cells, irregular grids with techniques such as quadtree Finkel and Bentley (1974); Knoll:06 or Voronoï diagram comprehensively discussed in the survey Aurenhammer (1991).

Our work is independent of the particular method used to obtain the discretization. We only work under the hypothesis that a feasible plan on the graph generated by the discretization is also feasible in the continuous space.

2.1 Topological graph

Compared to the graphs used in MAPP, we also consider communication edges which specify whether agents at two different locations can communicate. We call graphs with this additional information topological graphs. The formal definition is the following.

Definition 1** (Topological graph).**

A topological graph is a tuple $G=\langle V,\rightarrow,\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\rangle$, with $V$ a finite set of nodes containing a distinguished element $B$, $\rightarrow\subseteq V\times V$ a set of movement edges and $\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\subseteq V\times V$ a set of undirected communication edges.

The node $B$ symbolizes the supervision station from which the agents start the mission. A topological graph is undirected if $\langle V,\rightarrow\rangle$ is an undirected graph.

We will now consider three subclasses of interest.

In most situations, if an agent can move to a location in one step, it can also communicate with an agent at that location. This class has been discussed in Tateo et al. (2018). We call topological graphs satisfying this requirement neighbor-communicable. An example is given in Figure 2(b).

Definition 2** (Neighbor-Communicable topological graph).**

A neighbor-communicable topological graph is a topological graph such that $v\rightarrow v^{\prime}$ implies $v\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}v^{\prime}$.

Another class of graphs is that of sight-moveable and is the main one for which we give efficient algorithms. First, this class requires the movement edges to be undirected and reflexive. Second, whenever an agent can communicate with another node, then it can also move to that node while maintaining the communication. This intuitively means that we disallow communication through obstacles. The formal definition follows, and an example is depicted in Figure 2(c).

Definition 3** (Sight-Moveable topological graph).**

A sight-moveable topologic graph is an undirected neighbor-communicable topological graph in which for all $v\in V$, $v\rightarrow v$ and whenever $v\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}v^{\prime}$, there exists a sequence $\rho=\langle\rho_{1},\dots,\rho_{n}\rangle$ of nodes such that $v=\rho_{1}$, $v^{\prime}=\rho_{n}$, $v\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\rho_{i}$ and $\rho_{i}\rightarrow\rho_{i+1}$ for all $i\in\{1,n\}$.

Last, we define the complete-communication topological graphs which are simply sight-moveable topological graphs with a complete communication topology. This subclass can model the use of a hovering connected agent which allows a constant communication for all the agents in the area. An example of such a graph is depicted in Figure 2(d), and the formal definition is the following.

Definition 4** (Complete-Communication topological graph).**

A complete-communication topological graph is a sight-moveable topological graph such that $\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=V\times V$.

Observe that complete-communication graphs are reflexive, undirected, connected graphs with $\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=V\times V$.

2.2 Execution

A plan or execution, in MAPP, is a list of configurations which describes the placement of the agents during the mission. The formal definition of a configuration is the following.

Definition 5** (Configuration).**

A configuration $c$ of $n$ agents in a topological graph $G$ is an element of $V^{n}$ denoted $c=\langle c_{1},\dots,c_{n}\rangle$ in which $c_{i}$ is the location of the agent $i$ such that the graph $\langle V_{a},\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\cap V_{a}\times V_{a}\rangle$ is connected with $V_{a}=\{B,c_{1},\dots,c_{n}\}$. We extend our notation and denote $c\rightarrow c^{\prime}$ when $c_{i}\rightarrow c^{\prime}_{i}$ for all $0<i\leq n$.

Furthermore, in the literature, MAPP asks to associate an agent to a specific goal. However, given that we are interested in covering an area with a fleet of agents, the anonymity is useful to get more efficient plans.

Anonymity. In the rest of this paper, we consider the agents to be anonymous. In other words, a configuration $c$ is equivalent to a configuration $c^{\prime}$ iff $c$ is a reordering of $c^{\prime}$.

Moreover, an important notion in MAPP is the computation of collision-free plans. In the drone case, in which we are particularly interested, one can place drones at different heights to avoid collisions. Additionally, most drones are, nowadays, equipped with local collision avoidance systems.

Collisions. We do not deal with meet- or head-on-collisions of agents, i.e. we allow two agents to be located in a same node, and to move in opposite directions of an edge within a step.

An execution $e$ of length $\ell$ with $n$ agents in a graph $G$ is a sequence of configuration $\langle c^{1},\dots,c^{\ell}\rangle$ such that for $c^{i}\rightarrow c^{i+1}$ for all $0<i<\ell$. A covering execution $e=\langle c^{1},\dots c^{\ell}\rangle$ of length $\ell$ with $n$ agents in a graph $G$ is an execution such that $c^{1}=c^{\ell}=\langle B,\dots,B\rangle$ and for all $v\in V$, there exists $i\in\{1,\ldots,\ell\}$ with $v\in c^{i}$.

2.3 Decision problems

We define the MAPP problems, the Reachability problem along with its bounded version, bReachability, for the makespan optimization of the plan. In addition, we define the Coverage problem and the bounded coverage, bCoverage.

Definition 6** (Reachability).**

Given a topological graph $G$ and a configuration $c$, decide if there is an execution $\langle c^{1},\dots,c^{\ell}\rangle$ in $G$ such that $c^{1}=\langle B,\dots,B\rangle$ and $c^{\ell}=c$.

Definition 7** (Coverage).**

Given a topological graph $G$ and $n\in\mathds{N}$ written in unary, decide if there exists a covering execution with $n$ agents.

Definition 8** (bReachability).**

Given a topological graph $G$, configuration $c$ and $\ell\in\mathds{N}$ written in unary, decide if there is an execution $\langle c^{1},\dots,c^{\ell^{\prime}}\rangle$ in $G$ s.t. $\ell^{\prime}\leq\ell$ and $c^{\ell^{\prime}}=c$.

Definition 9** (bCoverage).**

Given a topological graph $G$, $n,\ell\in\mathds{N}$ written in unary, decide if there exists a covering execution of length $\ell^{\prime}$ such $\ell^{\prime}\leq\ell$.

We study the restrictions of the above problems to classes of topological graphs. We denote by ${B}_{c}$ , with $B$ one of the four above problems restricted to the class of topological graph $c$ ($c$ can either be $dir$ for directed, $nc$ for neighbor-communicable, $sm$ for sight-moveable or $cc$ for complete-communication topological graphs).

2.4 Known results

The complexity of the decision problem associated to the minimization of the makespan is known to be NP-hard since Ratner and Warmuth (1986). Throughout the study of MAPP, NP-hardness was shown to hold on planar graphs Yu (2016) and, later, on 2D grid graphs Banfi et al. (2017). Variants of MAPP have been studied such as the package-exchange robot-routing problem Ma et al. (2016) where the robots are anonymous but not the package they exchange, is shown to be NP-hard. A class of grid graphs was shown to be solvable in polynomial time Wang and Botea (2009).

The connected version of MAPP was introduced in Hollinger and Singh (2012), in which a topological graph discretizes the space and it is proved that the existence of a plan for the reachability of a configuration of agents in a bounded amount of steps is NP-hard:

Theorem 10**.**

${bReachability}$*  restricted to undirected topological graphs is NP-hard Hollinger and Singh (2012).*

In Tateo et al. (2018), it is shown that deciding the existence of a feasible plan is PSPACE-complete:

Theorem 11**.**

${Reachability}$*  restricted to undirected topological graphs is PSPACE-complete Tateo et al. (2018).*

Authors prove this result for graphs with self-loops and a base Tateo et al. (2018) as in our setting (see Discussion following Theorem 1). The only difference with our setting is that the agents start at a specific configuration in Tateo et al. (2018). Nevertheless, it can be shown easily that our problem is equivalent by duplicating the base and adding edges so that the agents reach the initial configuration at the second step.

In the rest of the paper, we study the upper bounds and the lower bounds complexity of the defined decision problems on the previously defined topological graphs. The following sections present our results, respectively, for the general case, the neighbor-communicable graphs, sight-moveable graphs, and complete-communication graphs.

3 Directed Topological Graphs

We start with the following upper bound which is obtained by a straightforward guess and check algorithm:

Proposition 12**.**

${bCoverage}_{dir}$* and ${bReachability}_{dir}$ are in NP.*

In both cases, we can guess and check a path of bounded length in NP since the input is encoded in unary.

We furthermore establish the following results:

Theorem 13**.**

${Coverage}_{dir}$* and ${Reachability}_{dir}$ are PSPACE-complete.*

The upper bounds are obtained by a straightforward NPSPACE algorithm that guesses an execution by keeping in memory the last configuration, and, for ${Coverage}_{dir}$ , the set of visited regions. We conclude with Savitch’s Theorem (NPSPACE=PSPACE)Savitch (1970).

The lower bound on ${Reachability}_{dir}$ was proven in Theorem 11. We now concentrate on ${Coverage}_{dir}$ .

Lemma 14**.**

${Coverage}_{dir}$* is PSPACE-hard.*

Proof.

The proof is by reduction from ${Reachability}_{dir}$ in which the base node has a self-loop. As noted in the remark following Theorem 11, this problem remains PSPACE-hard. We map an instance $(G,c)$ of ${Reachability}_{dir}$ to the instance $G^{\prime}$ of ${Coverage}_{dir}$ where $G^{\prime}$ is depicted in Fig. 3. Let $k$ denote the number of agents in the instance $(G,c)$. $G^{\prime}$ contains $G$ as a subgraph, plus fresh nodes $v_{1},\dots,v_{k}$ and $s_{1},\dots,s_{k}$. An agent can move from any node of $G$ to $v_{1}$ and back.

Node $s_{1}$ can communicate with the base $B$, and node $v_{k}$ can communicate with all nodes of $G^{\prime}$. Furthermore, we have the communication edges $(s_{i},s_{i+1})$ and $(v_{i},v_{i+1})$ for all $1\leq i\leq k-1$. Now we prove that the $k$ agents can progress to the configuration ($c_{1},\dots,c_{k}$) in $G$ if and only if there exists a covering execution in $G^{\prime}$.

($\Rightarrow$) If the agents are in the configuration ($c_{1},\dots,c_{k}$) then they can progress in one step to configuration ($s_{1},\dots,s_{k}$). Then, they have no choice but progress to the configuration ($v_{1},\dots,v_{k}$). Once in this configuration, the agent placed on the node $v_{k}$ communicates with the base and with all other agents. This agent stays at $v_{k}$. Meanwhile the agent placed on the node $v_{1}$ will visit all unvisited nodes of $G$ and come back to $v_{1}$ while keeping communication to the base through the agent placed on $v_{k}$. Meanwhile, agents placed on $v_{2},\dots,v_{k-1}$ come back to $B$. Finally, when all the nodes have been visited, both agents on $v_{1}$ and $v_{k}$ come back to $B$.

($\Leftarrow$) If there exists a covering execution of the whole graph $G^{\prime}$, it means all nodes have been visited. In particular, node $s_{k}$ has been visited and let us consider the first time $t_{s_{k}}$ when $s_{k}$ is visited. Time $t_{s_{k}}-1$ denotes the time just before $t_{s_{k}}$.

Fact 15**.**

At time $t_{s_{k}}-1$, no node $v_{i}$ and no node $s_{i}$ were visited.

Proof.

Suppose by contradiction that a node $v_{i}$ was visited by some agent before $t_{s_{k}}$, then the only possibility such an agent to communicate to the base is that there is also an agent at $v_{k}$ at time $t_{s_{k}}$. But then, it means that $s_{k}$ was visited strictly before $t_{s_{k}}$, leading to a contradiction. Thus, no node $v_{i}$ were visited at time $t_{s_{k}}$ (thus at time $t_{s_{k}}-1$).

As no node $v_{i}$ are visited before $t_{s_{k}}$, no node $s_{i}$ are visited before $t_{s_{k}}-1$. ∎

Fact 16**.**

At time $t_{s_{k}}-1$, the configuration is $\langle c_{1},\dots,c_{k}\rangle$.

Proof.

At time $t_{s_{k}}$, as the agent at $s_{k}$ needs to communicate with the base, the only possibility is that the configuration is $\langle s_{1},\dots,s_{k}\rangle$. Thus, the only possibility is that configuration is $\langle c_{1},\dots,c_{k}\rangle$. ∎

Facts 15 implies that the prefix from time 0 to time $t_{s_{k}}-1$ of the covering execution is an execution in $G$. Fact 16 implies that sub-execution reaches $\langle c_{1},\dots,c_{k}\rangle$. ∎

4 Neighbor-Communicable Topological Graphs

In this subsection, we show that our problems remain hard for neighbor-communicable graphs.

Theorem 17**.**

${Coverage}_{nc}$* is PSPACE-complete.*

Proof.

The upper bound is given by Theorem 13.

For the lower bound on ${Coverage}_{nc}$ , the reduction given in Figure 3 is not adapted for neighbor communicable graphs. Indeed, all nodes may be visited although $c_{1},\dots,c_{k}$ is not reached: $v_{1}$ and $v_{k}$ can be reached by two lines of agents connected to the base, making the coverage of the full graph possible. We nevertheless give a similar reduction by adapting the previous reduction.

The corrected construction is given in Figure 4. When configuration $\langle c_{1},...,c_{k}\rangle$ is reached, the agents go through a first layer of length $k+1$ in which the first agent can communicate with $B$. Then they go through another layer of length $k+1$ in which the $k^{th}$ agent can communicate with $B$. This way, it is mandatory that all agents move at the same time to visit $\langle v_{1},\dots,v_{k}\rangle$. Once the $k^{th}$ agent is at $v_{k}$, all agents can communicate with $B$ wherever they are, so they can visit remaining states in the copy of $G$. Now let us prove that $\langle c_{1},\dots,c_{k}\rangle$ is reachable in $G$ iff it is possible to cover all nodes in $G^{\prime}$.

($\Rightarrow$) If $\langle c_{1},\dots,c_{k}\rangle$ is reachable in $G$, then we extend the execution to reach $\langle v_{1},\dots,v_{k}\rangle)$ and by the same trick as in Figure 3, the agent that reaches $v_{1}$ visits all the remaining unvisited nodes in $G$. Thus, we extend the execution for covering all nodes in $G^{\prime}$.

$(\Leftarrow)$ Suppose all nodes are visited in $G^{\prime}$. In particular, $v_{1}$ and $v_{k}$ are visited. Let us consider the first moment $t_{v_{i}}$ when a node $v_{i}$ is visited.

Fact 18**.**

At the first moment, the configuration of the agents is $\langle v_{1},\dots,v_{k}\rangle$.

Proof.

Let us prove that there is an agent at $v_{k}$. Suppose that at that moment there is no agent at $v_{k}$. Due to the topological graph $G^{\prime}$, the agent at $v_{i}$ is disconnected from the base since nodes that communicate directly to $B$ are too far from $v_{i}$: indeed, the top $k+1$-grid is too long and, for $i=1$, the path on left between $v_{1}$ and the copy of $G$ is too long. Contradiction.

The agent at $v_{k}$ came from the unique $2k+2$-long path from $c_{k}$ to $v_{k}$. Actually, $k+1$ steps before - let us call this moment $t_{s_{k}}$, she was on $s_{k}$. But at that time, due to the topological graph, there are $k$ agents on the row containing $s_{k}$, otherwise the agent at $s_{k}$ would have been disconnected from the base (the bottom $k+1$-grid is too long).

So $k+1$ times later $t_{s_{k}}$, all the $k$ agents are at $\langle v_{1},\dots,v_{k}\rangle$. ∎

Taking Fact 18 as granted, we consider time $t$ that is $2k+2$ steps before and we clearly have the following fact.

Fact 19**.**

At time $t$, the configuration is $\langle c_{1},\dots,c_{k}\rangle$.

Moreover, the following fact holds.

Fact 20**.**

At time $t$, no node outside $G$ were visited.

Proof.

By contradiction, if some node outside $G$ were visited, it means that some agent went out the copy of $G$. By definition of $G^{\prime}$, it would mean that a node $v_{i}$ would have been visited, before time $t$, hence strictly before $t_{v_{i}}$. Contradiction. ∎

To sum up, the prefix of the execution from $\langle B,\dots,B\rangle$ to $\langle c_{1},\dots,c_{k}\rangle$ is fully inside the copy of $G$. So $\langle c_{1},\dots,c_{k}\rangle$ is reachable in $G$.

∎

5 Sight-Moveable Topological Graphs

In this subsection, we show that ${Reachability}_{sm}$ and ${Coverage}_{sm}$ are in LOGSPACE while the bounded version ${bReachability}_{sm}$ is NP-complete.

5.1 Upper bounds

The results in this subsection are based on a result of Reingold Reingold (2008), who proved that the problem of checking the connectivity of two nodes $s$ and $t$ in an undirected graph, namely USTCONN, is in LOGSPACE.

Theorem 21**.**

USTCONN  is in LOGSPACE Reingold (2008).

Proposition 22**.**

${Reachability}_{sm}$* is in LOGSPACE.*

Proof.

The idea of the proof is to reduce ${Reachability}_{sm}$ to UCONN, that is the problem of deciding whether an undirected graph is connected. From Theorem 21, we can reduce UCONN  to USTCONN  by simply looping over all pairs of nodes ($s,t)$ and checking for a path from $s$ to $t$. Therefore, UCONN  is in LOGSPACE.

Now we describe the logarithmic space reduction of ${Reachability}_{sm}$  to UCONN. Let $G=\langle V,\rightarrow,\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\rangle$ a sight-moveable topological graph and $c$ a configuration. Let $V^{\prime}=\{c_{1},\dots,c_{n},B\}$. The configuration $c$ is reachable iff the restriction of $\mathfrak{G}^{\prime}:=(V,\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$ to the nodes in $V^{\prime}$ is

-connected. Indeed, if it is, then $c$ is reachable: each agent follows some $\rightarrow$-path from $B$ to $c_{i}$ contained in a

-path from $B$ to $c_{i}$. In other words, $(G,c)$ is a positive ${Reachability}_{sm}$ -instance iff $\mathfrak{G}^{\prime}$ is a positive UCONN-instance. The reduction is in logarithmic space: we compute $\mathfrak{G}^{\prime}$ by enumerating all $(u,v)$

-edges in $G$, and we output $(u,v)$ when $u,v\in V^{\prime}$. We recall that we only take into account the working memory for computing $\mathfrak{G}^{\prime}$; the output – $\mathfrak{G}^{\prime}$ itself – is not taken into account in the used space (see e.g. Sipser (1997), Ch. 8, Def. 8.21). ∎

Proposition 23**.**

${Coverage}_{sm}$* is in LOGSPACE.*

Proof.

First we prove that the bounded version of the connectivity in undirected graphs is also in LOGSPACE.

Lemma 24**.**

Bounded-USTCONN, that is the problem, giving an undirected graph $\mathfrak{G}$, two nodes $s,t$, an integer $n$ written in binary, of deciding whether there is a path of length at most $n$ from $s$ to $t$ in $G$ is in LOGSPACE.

Proof.

We reduce Bounded-USTCONN  to USTCONN  in logarithmic space as follows. From a Bounded-USTCONN  instance $(\mathfrak{G},s,t,n)$ we construct in logarithmic space a USTCONN  instance $(\mathfrak{G}^{\prime},s^{\prime},t^{\prime})$:

1. The nodes of $\mathfrak{G}^{\prime}$ are pairs $(v,j)$ where $v$ is a node of $\mathfrak{G}$ and $j$ is an integer in $\{0,n\}$ but smaller than the number of nodes in $\mathfrak{G}^{\prime}$;

2. $\mathfrak{G}^{\prime}$contains an edge between $(v,j)$ and $(v^{\prime},j+1)$ when there is an edge between $v$ and $v^{\prime}$ in $\mathfrak{G}$ or when $v=v^{\prime}$;

3. $s^{\prime}=(s,0)$and $t^{\prime}=(t^{\prime},n)$.

∎

Now let $G=\langle V,\rightarrow,\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\rangle$ be a sight-moveable topologic graph and $n$ an integer written in binary. To test whether $(G,n)$ is a positive instance of Coverage, it suffices to check that there is a path from any node $v$ to the base $B$ with at most $n$ communication edges. To do that, we test sequentially, for all $v$, that $((V,\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}),v,B,n)$ is a positive instance of Bounded-USTCONN . Thus, we obtain an algorithm in logarithmic space to decide Coverage.

∎

5.2 Lower bounds

We now focus on the NP lower bound of ${bReachability}_{sm}$ .

Proposition 25**.**

${bReachability}_{sm}$* is NP-hard for a fixed execution length $\ell\geq 3$.*

Proof.

The proof is by polynomial time reduction from 3-SAT problem (see Karp (1972)). Given a 3-SAT instance, set of clauses $c_{1},\ldots,c_{m}$ with variables $x_{1},\ldots,x_{n}$, we describe the construction of an instance $(G,c)$ of ${bReachability}_{sm}$ with $k=n+m$ agents.

The topological graph $G=\langle V,\rightarrow,\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\rangle$ is constructed as follows. We start by placing the base $B$ from which the agents start their mission.

Please recall that a sight-moveable graph is also a neighbor-communicable graph so all movements edges are also communication edges in the construction below even if not explicitly stated.

For each variable ${x}$, we construct a gadget composed of 5 nodes connected to the base depicted in Figure 6(a): nodes ${x}$, $\neg{x}$, staging nodes $n_{x}$, $n_{\neg{x}}$ and a goal node $g_{x}$. We add movement edges from $B$ to $n_{x}$, from $n_{x}$ to ${x}$ and from ${x}$ to $g_{x}$ (resp. from $B$ to $n_{\neg{x}}$, from $n_{\neg{x}}$ to $\neg{x}$ and from $\neg{x}$ to $g_{x}$). As for the communication, the node ${x}$ (res. $\neg{x}$) communicates with the base.

For each clause $\mathsf{c}$, we construct a gadget composed of 3 nodes depicted in Figure 6(b). We create a node $\mathsf{c}$, a staging node $n_{\mathsf{c}}$ and a goal node $g_{\mathsf{c}}$. We add movement edges from $B$ to $n_{c}$, from $n_{\mathsf{c}}$ to $\mathsf{c}$ and from $\mathsf{c}$ to $g_{\mathsf{c}}$. The communication between a clause $\mathsf{c}$ and a literal ${x}$ or $\neg{x}$ is dictated by the existence of the literal in the clause: $\mathsf{c}_{i}\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{x}_{j}$ if and only if ${x}_{j}\in\mathsf{c}_{i}$; and $\mathsf{c}_{i}\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\lnot{x}_{j}$ if and only if $\lnot{x}_{j}\in\mathsf{c}_{i}$.

We add movement edges from $g_{{x}_{i}}$ to $g_{{x}_{i+1}}$, and from $g_{\mathsf{c}_{i}}$ to $g_{\mathsf{c}_{i+1}}$ for all $1\leq i<n$, as well as we from $g_{{x}_{n}}$ to $g_{\mathsf{c}_{1}}$. Last, we add a fully connected path containing 3 fresh nodes from $g_{{x}_{1}}$ to the base such that $g_{{x}_{1}}\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}B$, in the sense that all nodes of this path have communication edges between them. This translation is polynomial in the number of clauses and variables. The construction is depicted in Figure 5. The snake-like path from $g_{{x}_{1}}$ to $B$ is the fully connected path.

From a 3-SAT instance, one can construct the graph $G$ and ask for an execution of length 3 to reach the configuration $\langle g_{{x}_{1}},\dots,g_{{x}_{n}},g_{\mathsf{c}_{1}},\dots,g_{\mathsf{c}_{m}}\rangle$.

Fact 26**.**

$G$* is a sight-moveable topological graph.*

Proof.

One can see that the single communication edges created by the construction, apart from the ones induced by the movement, are the communication between the base $B$ and the nodes ${x}_{i}$ and $\neg{x}_{i}$. Hence, a path does exist under the communication of $B$ to reach ${x}_{i}$. ∎

Now let us prove that a 3-SAT instance is satisfiable iff there exists an execution of at most 3 steps in the graph $G$.

($\Rightarrow$) We show that if a 3-SAT instance is satisfiable then there exists an execution of at most 3 steps in the graph $G$ built from it. Let $val$ be a truth assignment which satisfies the instance. Recall that there are $n+m$ agents. The first step of the execution consists in moving an agent in each $n_{c_{i}}$, and for each variable $x_{j}$, moving one agent to $n_{x_{j}}$ if the $val(x_{j})=1$ and to $n_{\lnot x_{j}}$ otherwise. Note that all staging nodes communicate with $B$ since the graph is neighbor-communicable.

In the second step, all agents progress to their unique successors other than $B$. While all nodes $x_{j}$ and $\lnot x_{j}$ are connected to $B$, a node $c_{i}$ is connected to $B$ if and only if there is an agent in one of its literals. This is the case since $val$ satisfies the formula. In the third step of the execution, agents go to states $g_{x_{j}}$ and $g_{c_{i}}$. Here, the connection with the base is ensured since $g_{x_{1}}$ is connected to it, and $g_{x_{2}}$ is connected to $g_{x_{1}}$, $g_{x_{3}}$ is connected to $g_{x_{2}}$ and so on.

This execution is thus a solution of ${bReachability}_{sm}$ with bound $\ell=3$.

($\Leftarrow$) We now show that if there exists an execution of at most 3 steps in the graph $G$ constructed from a 3-SAT instance, then the instance is satisfiable. Assume we have an execution $e$ of at most 3 steps with the last configuration being $\langle g_{{x}_{1}},\dots,g_{{x}_{n}},g_{\mathsf{c}_{1}},\dots,g_{\mathsf{c}_{m}}\rangle$.

The only shortest path from $B$ to $g_{c_{i}}$ is of length $3$ and goes through $n_{c_{i}}$. For states $g_{x_{j}}$, the only shortest paths are also of length $3$ and go through either $n_{x_{j}}$ or $n_{\lnot x_{j}}$. Thus, in order to reach the given target configuration, at the initial step, agents must cover the states $n_{c_{i}}$ and either $n_{x_{j}}$ or $n_{\lnot x_{j}}$ for all $i,j$. At the second step, following the above mentioned shortest paths, agents will be at states $c_{i}$ and either $x_{j}$ or $\lnot x_{j}$ depending on the staging nodes they were occupying. The last step is the target configuration. Since the agents are connected at the second, it follows that for each clause $c_{i}$, the state corresponding to some literal of $c_{i}$ is occupied by an agent. Thus the valuation on variables encoded by the choices of the agents satisfies the 3-SAT instance.

∎

From Propositions 12 and 25, we have:

Theorem 27**.**

${bReachability}_{sm}$* is NP-complete.*

6 Complete-Communication Topological Graphs

The following result relies on the fact that the communication constraints are trivial this class.

Proposition 28**.**

${bReachability}_{cc}$* is in LOGSPACE.*

Proof.

From Lemma 24, one can construct an algorithm in LOGSPACE for ${bReachability}_{cc}$ . Indeed, given a configuration $c$ and $\ell\in\mathds{N}$, the straightforward iteration on the locations $c_{i}$ followed by the verification of a path of at most $\ell$ (given in unary) steps from $B$ to $c_{i}$ yields a sound and complete algorithm for ${bReachability}_{cc}$ . ∎

Our NP lower bound proof of the ${bCoverage}_{cc}$ problem is by reduction from the grid Hamiltonian cycle (G-HC) problem which is the Hamiltonian cycle problem restricted to grid graphs and is NP-complete Itai et al. (1982).

Theorem 29**.**

${bCoverage}_{cc}$* is NP-complete.*

Proof.

The upper bound follows from Proposition 12.

We give a polynomial-time reduction from the G-HC problem. Consider a graph $G=\langle V,E\rangle$, an instance of G-HC.

Consider the sight-moveable topological graph $G^{\prime}=\langle V,\rightarrow,\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\rangle$ with $\rightarrow~{}=E$ and $\scalebox{0.5}{ \leavevmode\hbox to27.52pt{\vbox to7.03pt{\pgfpicture\makeatletter\hbox{\hskip 3.33301pt\lower-5.60927pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@setlinewidth{2.84528pt}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{-2.27626pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=V\times V$ and associate a single agent and the bound $|V|$ to the ${bCoverage}_{cc}$ instance. We call a simple cycle containing all vertices a tour. We prove that there exists a tour $t$ in $G$ iff there exists a covering execution of length $|V|$ in $G^{\prime}$.

($\Rightarrow$) Any tour of $G$ is a valid execution satisfying ${bCoverage}_{cc}$ since the communication edges form a complete graph, and the bound is $|V|$.

($\Leftarrow$) Let us suppose that we have an execution of length $|V|$ which covers the graph $G^{\prime}$. The execution starts and ends at $B$ and visits all nodes in $|V|$ steps. Hence, the execution visits all nodes only once and is a cycle in the graph.

∎

7 Related Work

The coverage planning is an interesting approach to path planning. Indeed, a covering plan can be used for fields such as floor cleaning, lawn mowing, etc. A survey of this field appears in Choset (2001). This multi-agent extension has the ability to reduce the length of the overall mission and also reach parts of the area a single agent would not able to. This problem was studied in Rekleitis et al. (1997) for two agents. As shown in the survey by Chen et al. Chen et al. (2014), many coverage problems have been addressed by using analytic techniques. For instance, in Yanmaz (2012) and Teacy et al. (2010), they consider UAVs that should cover an area while staying connected to the base, but only empirically study some path planning algorithms without proving their algorithms formally.

We advocate formal methods that give formal guarantees and have already been applied to generate plans for robots and UAVs. Model checking has been applied to robot planning (see Lacerda et al. (2014)) and to UAVs Webster et al. (2011). Humphrey Humphrey (2013) shows how to use LTL (linear-temporal logic) model checking for capturing response and fairness properties in cooperation (for instance, if a task is requested then it is eventually performed).

Bodin et al. Bodin et al. (2018) treat a similar problem except that the UAVs cover the graph without returning to the base. Without the return-to-the-base constraint, we claim that all our hardness results still hold, except for ${bCoverage}_{cc}$ . They provide an implementation by describing the problem in Planning Domain Description Language and then run the planner Functional Strips Francès et al. (2017).

Murano et al. Murano et al. (2015) advocate for a graph-theoretic representations of states, that is, by assigning locations to agents as in Definition 5. In Aminof et al. (2016); Rubin (2015), a general formalism is given to specify LTL and monadic second-order logic properties, which are expressive enough to describe the connectivity constraint. They provide an algorithm for parametrized verification in the sense that they check a temporal property in a class of graphs. This is relevant for partially-known environments. The algorithm described is non-elementary (i.e. the running time cannot bounded by any tower of exponentials) and therefore not usable in practice. We believe that this is an important problem and our paper identifies an efficient and relevant fragment.

The multiple traveling salesman problem (mTSP) is a generalization of the traveling salesman problem (TSP) in which multiple salesmen are located at a depot Anbuudayasankar et al. (2016). mTSP asks for the coverage of all cities so as to minimize the total plan cost by visiting each city exactly once. An overview of TSP and its extensions are presented Matai et al. (2010). The ${Coverage}$  problem is related to mTSP, since we use results on Hamiltonian cycle to prove the NP-hardness of ${bCoverage}_{cc}$ . However, we wish to minimize the length of the execution and not the cost of the execution. Those problems are equivalent on unit graphs, but it is not trivial to use general results on mTSP in order to solve ${Coverage}$ . Furthermore, to the best of our knowledge, connected versions of mTSP and VRP have not been studied.

8 Conclusion

Sight-moveable topological graphs we introduced in this work only constrain the communication graph. One can be interested to constrain the movement graph to a planar graph or a 2D grid given the common usage of grid modelling of the environment. Given the intractability of MAPP on planar graphs Yu (2016) and on general 2D grid graphs Banfi et al. (2017), it is likely that this problem is intractable as well. Furthermore, in Tateo et al. (2018), the decision is proved to stay PSPACE-complete on planar graphs and grids as well. However, one can study this problem on solid grid graphs, given that the Hamiltonian cycle is tractable on such graphs Umans and Lenhart (1997).

One can note that our NP lower bound reductions hold without the anonymity of the agents. Indeed, the ${bCoverage}$  case is straightforward and for ${bReachability}$  case, each agent can be associated to a clause or variable, so the reduction would still hold.

We do not know if ${Coverage}$  remains hard when the $\rightarrow$-relations become symmetric, depicted in Figure 7 as a question mark. We think this open issue is important since symmetric $\rightarrow$-relations (if UAVs can go from $v$ to $v^{\prime}$, they can also come back from $v^{\prime}$ to $v$) are relevant for practical applications. We plan to study the parametrized complexity Downey and Fellows (1999) of our problems - parameters could be the treewidth of the topological graph, the number of UAVs.