We study the problem of collective tree exploration (CTE) where a team of $k$ agents is tasked to traverse all the edges of an unknown tree as fast as possible, assuming complete communication between the agents. In this paper, we present an algorithm performing collective tree exploration in only $2n/k+O(kD)$ rounds, where $n$ is the number of nodes in the tree, and $D$ is the tree depth. This leads to a competitive ratio of $O(\sqrt{k})$ for collective tree exploration, the first polynomial improvement over the initial $O(k/\log(k))$ ratio of [FGKP06]. Our analysis relies on a game with robots at the leaves of a continuously growing tree, which is presented in a similar manner as the `evolving tree game' of [BCR22], though its analysis and applications differ significantly. This game extends the `tree-mining game' (TM) of [Cos23] and leads to guarantees for an asynchronous extension of collective tree exploration (ACTE). Another surprising consequence of our results is the existence of algorithms $\{A_k\}_{k\in \mathbb{N}}$ for layered tree traversal (LTT) with cost at most $2L/k+O(kD)$, where $L$ is the sum of edge lengths and $D$ is the tree depth. For the case of layered trees of width $w$ and unit edge lengths, our guarantee is thus in $O(\sqrt{w}D)$.

翻译：我们研究集体树探索问题，其中由$k$个智能体组成的团队需在完全通信条件下，尽可能快地遍历一棵未知树的所有边。本文提出一种只需$2n/k+O(kD)$轮即可完成集体树探索的算法，其中$n$为树节点数，$D$为树深度。该算法实现了$O(\sqrt{k})$的竞争比，这是自[FGKP06]中原始$O(k/\log(k))$比值以来的首次多项式改进。我们的分析依赖于一个在持续生长的树叶子节点处进行机器人博弈的模型，其呈现方式与[BCR22]的"演化树博弈"类似，但分析与应用存在显著差异。该博弈扩展了[Cos23]的"树挖掘博弈"，并为集体树探索的异步扩展提供了性能保证。另一个令人意外的结果是存在算法族$\{A_k\}_{k\in \mathbb{N}}$用于分层树遍历，其代价不超过$2L/k+O(kD)$，其中$L$为边长度之和，$D$为树深度。对于宽度为$w$且边长为单位长度的分层树，我们的保证为$O(\sqrt{w}D)$。