AI 中文总结
本文针对最短公共超串问题，证明贪心算法对长度≥6的字符串近似比达2，且确定长度为3的字符串的近似比为9/5，推进了悬而未决40年的贪心超串猜想研究。
AI 中文摘要
在最短公共超串（Shortest Common Superstring, SCS）问题中，给定一组字符串，要求找到一个包含所有输入字符串作为子串的最短字符串。贪心超串猜想指出，自然贪心算法（每次合并一对重叠度最大的字符串）的近似比为2，该算法运行时间为线性，可说是SCS问题最简单的近似算法；若猜想成立，其近似保证将优于已知最优算法。该猜想已悬而未决40年，且所有字符串长度为k的实例的近似比ρ_k也未知，对k≥3，有2−1/k≤ρ_k≤min{(k+1)/2,3.396}。本文证明长度为6的字符串已足以达到近似比2，即对所有k≥6，ρ_k≥2；还证明ρ_3=9/5，完全确定了贪心算法对长度为3的字符串的最坏情况行为。
英文摘要
In the Shortest Common Superstring (SCS) problem, one is given a set of strings and is asked to find a string of minimum length containing each of the input strings as a substring. The greedy superstring conjecture states that the following natural greedy algorithm has approximation ratio $2$: while there is more than one string, select the pair of strings with the maximum overlap, merge them, and add the merged string back to the set. The greedy algorithm works in linear time and is probably the simplest possible approximation algorithm for SCS. If the conjecture holds, then the greedy algorithm also surpasses the approximation guarantees of the best known approximation algorithms. The conjecture is open for $40$ years already and even the approximation ratio $ρ_k$ in the special case in which input strings have length $k$ has not yet been found: for all $k \ge 3$, $2-1/k \le ρ_k \le \min\{(k+1)/2, 3.396\}$. We prove that already for strings of length $6$, the approximation ratio of the greedy algorithm is at least $2$: $ρ_k \ge 2$ for all $k \ge 6$. We also show that $ρ_3=9/5$, thus completely characterizing the worst-case behavior of the greedy algorithm for strings of length $3$.
CommentsFix typos (mostly in section 3)
