The Greedy Superstring Algorithm Achieves Ratio 2 for Strings of Length 6 Already

2026-08-20

AI Summary

研究证明了长度为6的字符串已能使贪婪算法达到2的近似比，解决了最短公共超字符串问题中的贪婪超字符串猜想部分问题。

Abstract

In the Shortest Common Superstring (SCS) problem, one is given a set of strings and asked to find a shortest string containing every input string as a substring. The greedy superstring conjecture states that the natural greedy algorithm, which repeatedly merges a pair of strings with maximum overlap, has approximation ratio $2$. The greedy algorithm runs in linear time and is arguably the simplest approximation algorithm for SCS. If the conjecture holds, it would also surpass the approximation guarantees of the best known algorithms. The conjecture has remained open for 40 years. Even the approximation ratio $ρ_k$ for instances whose strings all have length $k$ is unknown; for every $k \ge 3$, we have $2 - 1/k \le ρ_k \le \min\{(k+1)/2, 3.396\}$. We prove that strings of length 6 already suffice to achieve approximation ratio $2$: $ρ_k \ge 2$ for every $k \ge 6$. We also prove that $ρ_3 = 9/5$, completely characterizing the worst-case behavior of the greedy algorithm for strings of length 3.

Problem

• Shortest Common Superstring
• greedy algorithm
• approximation ratio

Innovation

• Greedy Superstring Algorithm
• Approximation Ratio
• Shortest Common Superstring (SCS)
• Strings of Length 6