An Optimal Algorithm for Binary Closest String

2026-05-29

AI Summary

This work addresses the Binary Closest String problem: given a set of binary strings of equal length, find a string that minimizes the maximum Hamming distance to all strings in the set. The paper proposes a concise and efficient randomized parameterized algorithm, using the optimal distance \(d\) as the parameter, which improves the best-known running time from \(O^*(5^d)\) to \(O^*(4^d)\). This result matches the established fine-grained complexity lower bound, thereby achieving a dual advance in both theoretical efficiency and algorithmic simplicity.

Abstract

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.

Problem

• Binary Closest String
• Hamming distance
• parameterized algorithm
• optimal distance

Innovation

• Binary Closest String
• parameterized algorithm
• Hamming distance
• fine-grained complexity
• randomized algorithm