Tackling the Minimal Superpermutation Problem

Abstract

A superpermutation on $n$ symbols is a string that contains each of the $n!$ permutations of the $n$ symbols as a contiguous substring. The shortest superpermutation on $n$ symbols was conjectured to have length $\sum_{i=1}^n i!$. The conjecture had been verified for $n \leq 5$. We disprove it by exhibiting an explicit counterexample for $n=6$. This counterexample was found by encoding the problem as an instance of the (asymmetric) Traveling Salesman Problem, and searching for a solution using a powerful heuristic solver.