We study deterministic online algorithms for the problem of chasing sets of cardinality at most $k$ in a metric space, also known as metrical service systems and equivalent to width-$k$ layered graph traversal. We resolve the 30-year-old gap of $Ω(2^k)\cap O(k2^k)$ on the competitive ratio of this problem by giving an $O(2^k)$-competitive deterministic algorithm. This bound is optimal even among randomized algorithms against adaptive adversaries. We also (slightly) improve the deterministic lower bound to $D_k$, defined recursively by $D_1=1$ and $D_{k+1}=2D_k+\sqrt{8+8D_k}+3$, which we conjecture to be exactly tight. For $k=3$, we provide a matching upper bound of $D_3$. Our results imply slightly improved upper and lower bounds for distributed asynchronous collective tree exploration and for the $k$-taxi problem, respectively. Our algorithm generalizes the classical doubling strategy, previously known to be optimal for $k=2$. The previous best bound for general $k$ was achieved by the generalized work function algorithm (WFA), and was known to be tight for WFA. Our improved bound therefore implies that WFA is sub-optimal for chasing small sets.

翻译：我们研究度量空间中基数至多为$k$的集合追逐问题的确定性在线算法（该问题亦称度量服务系统，等价于宽度为$k$的层次图遍历）。通过给出$O(2^k)$竞争比的确定性算法，我们解决了该问题竞争比长达30年的$Ω(2^k)\cap O(k2^k)$差距。该界即使对对抗自适应对手的随机算法而言也是最优的。我们还（略微）将确定性下界改进为$D_k$，其递归定义为$D_1=1$且$D_{k+1}=2D_k+\sqrt{8+8D_k}+3$，我们猜想该界是精确紧的。对于$k=3$，我们给出了匹配的上界$D_3$。我们的结果分别为分布式异步集体树探索问题和$k$-出租车问题提供了略微改进的上界和下界。该算法推广了经典的加倍策略——该策略此前已知对$k=2$是最优的。此前一般$k$的最佳界由广义工作函数算法（WFA）实现，且已知该界对WFA是紧的。因此，我们改进的界表明WFA在追逐小集合问题上非最优。