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 $O(\mathsf{vc})$.

翻译：我们首次研究了集体图探索（Collective Graph Exploration，简称CGE）问题的参数化复杂度。在CGE问题中，输入包括一个无向连通图$G$和一个由$k$个机器人组成的集合，这些机器人初始时均位于$G$的同一顶点$r$处，且每个机器人拥有能量预算$B$。目标是判定$G$是否能在$B$个时间步内被这$k$个机器人探索完成，即：在$G$中存在$k$条闭迹（每条闭迹对应一个机器人），使得每条边至少被一条闭迹覆盖、每条闭迹均以顶点$r$为起点和终点，且任意闭迹的最大长度不超过$B$。遗憾的是，该问题即使在树上也是\textsf{NP-困难的}[Fraigniaud等人，2006]。进一步，我们证明即使对于树深度为3的树，该问题在参数$k$下仍是\textsf{W[1]-困难的}。鉴于该问题在树深度参数下具有\textsf{para-NP-困难性}，并受现实场景启发，我们研究了该问题在图顶点覆盖数（$\mathsf{vc}$）参数化下的复杂度，并证明该问题在参数$\mathsf{vc}$下是固定参数可处理的（\textsf{FPT}）。此外，我们还研究了CGE的优化版本（即优化$B$），并设计了一个具有$O(\mathsf{vc})$加法近似因子的近似算法。