We investigate the problem of collaborative tree exploration with complete communication introduced by [FGKP06], in which a group of $k$ agents is assigned to collectively go through all edges of an unknown tree in an efficient manner and then return to the origin. The agents have unrestricted communication and computation capabilities. The algorithm's runtime is typically compared to the cost of offline traversal, which is at least $\max\{2n/k,2D\}$ where $n$ is the number of nodes and $D$ is the tree depth. Since its introduction, two types of guarantee have emerged on the topic: the first is of the form $r(k)(n/k+D)$, where $r(k)$ is called the competitive ratio, and the other is of the form $2n/k+f(k,D)$, where $f(k,D)$ is called the competitive overhead. In this paper, we present the first algorithm with linear-in-$D$ competitive overhead, thereby reconciling both approaches. Specifically, our bound is in $2n/k + O(k^{\log_2 k} D)$ and thus leads to a competitive ratio in $O(k/\exp(0.8\sqrt{\ln k}))$. This is the first improvement over the $O(k/\ln k)$-competitive algorithm known since the introduction of the problem in 2004. Our algorithm is obtained for an asynchronous generalization of collective tree exploration (ACTE). It is an instance of a general class of locally-greedy exploration algorithms that we define. We show that the additive overhead analysis of locally-greedy algorithms can be seen through the lens of a 2-player game that we call the tree-mining game and that could be of independent interest.

翻译：我们研究了由[FGKP06]引入的具有完全通信能力的协作树探索问题，在该问题中，一组$k$个智能体被指派以高效方式共同遍历未知树的所有边，然后返回原点。智能体拥有不受限制的通信与计算能力。算法运行时间通常与离线遍历的成本进行比较，后者至少为$\max\{2n/k,2D\}$，其中$n$为节点数，$D$为树深度。自问题提出以来，学术界出现了两种性能保证形式：第一种为$r(k)(n/k+D)$，其中$r(k)$称为竞争比；另一种为$2n/k+f(k,D)$，其中$f(k,D)$称为竞争开销。本文首次提出具有线性$D$竞争开销的算法，从而调和了这两种方法。具体而言，我们的界为$2n/k + O(k^{\log_2 k} D)$，进而导出$O(k/\exp(0.8\sqrt{\ln k}))$的竞争比。这是自2004年问题提出以来，首次在$O(k/\ln k)$竞争比的算法上取得改进。该算法是基于集体树探索的异步推广（ACTE）获得的，属于我们定义的一类局部贪婪探索算法。我们证明，局部贪婪算法的加性开销分析可通过一个称为"树挖掘博弈"的双人博弈视角进行理解，该博弈本身可能具有独立的研究价值。