Collective Graph Exploration Parameterized by Vertex Cover

TL;DR

This paper studies the computational complexity of the Collective Graph Exploration problem, proving its hardness in general and trees, but also showing it is fixed-parameter tractable when parameterized by vertex cover number, and providing an approximation algorithm.

Contribution

The paper establishes the parameterized complexity of CGE with respect to vertex cover and introduces an approximation algorithm based on this parameter.

Findings

CGE is NP-hard even on trees.

CGE remains W[1]-hard when parameterized by k on trees of treedepth 3.

CGE is fixed-parameter tractable when parameterized by vertex cover number.

Abstract

We initiate the study of the parameterized complexity of the {\sc Collective Graph Exploration} ({\sc CGE}) problem. In {\sc CGE}, the input consists of an undirected connected graph $G$ and a collection of $k$ robots, initially placed at the same vertex $r$ of $G$, and each one of them has an energy budget of $B$. The objective is to decide whether $G$ can be \emph{explored} by the $k$ robots in $B$ time steps, i.e., there exist $k$ closed walks in $G$, one corresponding to each robot, such that every edge is covered by at least one walk, every walk starts and ends at the vertex $r$, and the maximum length of any walk is at most $B$. Unfortunately, this problem is \textsf{NP}-hard even on trees [Fraigniaud {\em et~al.}, 2006]. Further, we prove that the problem remains \textsf{W[1]}-hard parameterized by $k$ even for trees of treedepth $3$. Due to the \textsf{para-NP}-hardness of the problem parameterized by treedepth, and motivated by real-world scenarios, we study the parameterized complexity of the problem parameterized by the vertex cover number ($\mathsf{vc}$) of the graph, and prove that the problem is fixed-parameter tractable (\textsf{FPT}) parameterized by $\mathsf{vc}$. Additionally, we study the optimization version of {\sc CGE}, where we want to optimize $B$, and design an approximation algorithm with an additive approximation factor of $O(\mathsf{vc})$.