The Greedy Superstring Conjecture Is False
Nikolai Chukhin, Rishikesh Gajjala, Juha Harviainen, Alexander S. Kulikov, Ivan Mihajlin, Hiroki Shibata, Alexander V. Smal
原文摘要
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. Since the problem is NP-hard, various approximation algorithms have been proposed for it. The currently best known approximation ratio is $2.466$ due to Englert, Matsakis, and Veselý; a better approximation ratio $2$ has been announced by OpenAI just recently. In 1988, Tarhio and Ukkonen introduced a simple greedy algorithm that iteratively takes two strings with the longest overlap and merges them together. They showed that the greedy algorithm may produce a superstring that is about two times longer than the optimal one and conjectured that this bound is tight. This has become known as the Greedy Superstring Conjecture and has remained open for almost four decades. The best known upper bound on the approximation ratio of the greedy algorithm is $3.396$ due to Englert, Matsakis, and Veselý. In this paper, we present a series of results tightening the bounds of the greedy algorithm, with the main contributions being as follows: 1. We refute the Greedy Conjecture by constructing a family of instances for which a lower bound on the approximation ratio tends to $2.4$. 2. We prove that the greedy algorithm is $3$-approximate. 3. For input strings of fixed length $k$, we prove that the approximation guarantee is exactly $9/5$ for $k=3$ and at least $2$ for all $k \ge 6$.
cs.DS