We revisit the Binary Closest String problem, which asks, given a set of binary strings $X \subseteq \{0, 1\}^n$, to compute a string minimizing the maximum Hamming distance to $X$. A long line of work has focused on parameterized algorithms with respect to the optimal distance $d$, yielding a sequence of improvements from $O^*(d^d)$ through $O^*(16^d)$, $O^*(9.513^d)$, $O^*(8^d)$, $O^*(6.731^d)$ to the current best-known running time of $O^*(5^d)$ [Chen, Ma, Wang; Algorithmica '16]. We present a faster randomized algorithm running in time $O^*(4^d)$. Our result matches a recent fine-grained lower bound [Abboud, Fischer, Goldenberg, Karthik C.S., Safier; ESA '23], and is therefore conditionally optimal. As an extra benefit, our algorithm is remarkably simple.

翻译：我们重新审视二元最近字符串问题：给定一组二进制字符串 $X \subseteq \{0, 1\}^n$，要求计算一个使得到 $X$ 的最大汉明距离最小化的字符串。长期以来，该问题的研究集中在基于最优距离 $d$ 的参数化算法上，并取得了一系列改进：从 $O^*(d^d)$ 到 $O^*(16^d)$、$O^*(9.513^d)$、$O^*(8^d)$、$O^*(6.731^d)$，直至当前已知最佳运行时间 $O^*(5^d)$ [Chen, Ma, Wang; Algorithmica '16]。我们提出了一种更快的随机算法，其运行时间为 $O^*(4^d)$。该结果与近期基于细粒度归约的下界 [Abboud, Fischer, Goldenberg, Karthik C.S., Safier; ESA '23] 相吻合，因此在条件假设下是最优的。作为额外优势，我们的算法极为简洁。