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

Abstract

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 $\rho_k$ in the special case in which input strings have length $k$ has not yet been found: for all $k \ge 3$, $2-1/k \le \rho_k \le \min\{(k+1)/2, 3.396\}$. We prove that already for strings of length $6$, the approximation ratio of the greedy algorithm is at least $2$: $\rho_k \ge 2$ for all $k \ge 6$. We also show that $\rho_3=9/5$, thus completely characterizing the worst-case behavior of the greedy algorithm for strings of length $3$.