A searcher faces a graph with edge lengths and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge length. The goal is to minimize the weighted sum of the exploration times over all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the general case, we give a $(2\mathrm{e}+\varepsilon)$-approximation for any $\varepsilon > 0$. For the case that all vertices have unit weight, we provide a $2\mathrm{e}$-approximation. Finally, we provide a PTAS for the case of a Euclidean graph. Previously, for all cases only an $8$-approximation was known.

翻译：搜索者面对一个具有边长和顶点权重的图，初始时仅探索了给定的起始顶点。在每一步中，搜索者向解中添加一条连接未探索顶点与已探索顶点的边。这需要的时间等于边长。目标是所有顶点上的探索时间加权和最小化。我们证明该问题难以近似，并提供了具有改进近似保证的算法。对于一般情况，我们给出任意 $\varepsilon > 0$ 下的 $(2\mathrm{e}+\varepsilon)$-近似。对于所有顶点具有单位权重的情况，我们给出 $2\mathrm{e}$-近似。最后，针对欧几里得图的情况，我们提供了一种PTAS。此前，所有情况仅有8-近似已知。