最短公共超串问题的紧环覆盖不等式
A Tight Cycle-Cover Inequality for Shortest Common Superstring
浏览论文内容
中文总结 AI 辅助
本研究针对最短公共超串问题，通过改进重叠图环覆盖不等式，将SCS近似比提升至7/3，贪心算法保证改进至3，并证明新上界不可再优化。
中文摘要 AI 辅助
在最短公共超串问题（SCS）中，给定一个有限字符串集合，要求找到一个最短的字符串，使得每个输入字符串都作为其子串出现。其已知的最佳近似比为 $2.466$，而目前最大重叠贪心算法的近似保证的最强上界为 $3.396$（Englert, Matsakis, 和 Veselý, 2023），尽管推测该上界为 $2$。我们改进了这两个近似保证：SCS 具有 $\frac{7}{3}$ 近似比，且贪心算法的近似保证至多为 $3$。我们证明的主要技术要素是与输入字符串相关的重叠图的最小费用环覆盖的某个不等式。此前对贪心算法最坏情况保证的每次改进以及近期针对一般 SCS 的两个创纪录保证均由该不等式驱动。我们通过将该不等式推至极限来改进它：针对该不等式的特定系数，我们给出了一个新的上界，并证明该上界无法进一步改进。
英文摘要
In the Shortest Common Superstring problem (SCS), one is given a finite set of strings and is asked to find a shortest string containing every input string as a substring. Its best known approximation ratio is $2.466$, whereas the currently strongest upper bound on the approximation guarantee of the maximum-overlap greedy algorithm is $3.396$ (Englert, Matsakis, and Vesel{ý}, 2023), though it is conjectured to be $2$. We improve both approximation guarantees: SCS admits a $\frac{7}{3}$ approximation and the approximation guarantee of the greedy algorithm is at most $3$. The main technical ingredient of our proof is a certain inequality for optimum cycle covers of an overlap graph associated with the input strings. Every previous improvement of greedy's worst-case guarantee and the two recent record guarantees for general SCS are driven by it. We improve this inequality by pushing it to its limit: for a particular coefficient of this inequality, we show a new upper bound and prove that it cannot be improved further.
发表机构
- JetBrains Research（JetBrains研究院）
机构由 AI 辅助整理，请以论文原文为准。
