Tackling the Minimal Superpermutation Problem (1408.5108v1)

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.

• The paper disproves the minimal-length conjecture by finding a six-symbol superpermutation of length 872 instead of the conjectured 873.
• The methodology converts permutations into a directed graph and uses heuristic solvers like Concorde and LKH to approximate a minimal Hamiltonian path.
• The findings challenge traditional combinatorial assumptions and highlight the power of computational heuristics in addressing NP-hard problems.

Understanding the Minimal Superpermutation Problem

Robin Houston's paper, "Tackling the Minimal Superpermutation Problem," explores a combinatorial problem that involves finding the shortest string, or superpermutation, that contains all permutations of a set of $n$ symbols as contiguous substrings. Historically, it was conjectured that the minimum length of such a superpermutation is given by $\sum_{i=1}^n i!$, but this conjecture had only been verified up to $n = 5$.

Key Contributions

The author's primary contribution is the disproval of this minimal-length conjecture for $n = 6$. By employing a computational approach that reconfigures the problem into an instance of the asymmetric Traveling Salesman Problem (TSP), the research successfully identifies a superpermutation of length 872 for six symbols, shorter than the conjectured length of 873. The strategy utilizes heuristic solvers like Concorde and LKH to approximate solutions effectively, even when exact solutions remain computationally infeasible.

Methodological Insights

The approach converts the superpermutation problem into a directed graph, representing each permutation as a vertex and connecting vertices with edge weights determined by shared sequences. The challenge is akin to locating a Hamiltonian path that forms a minimal-weight circuit. Concorde was employed to tackle the symmetric version of TSP by converting the asymmetric problem using the Jonker-Volgenant transformation. However, the asymmetric support in the LKH solver allowed for a direct approach, successfully finding a viable solution.

Houston's work exemplifies the broader applicability of NP-hard problem solvers in tackling complex combinatorial issues, which has often been overlooked within pure mathematics until recently.

Implications and Future Directions

Practically, the paper showcases the power of heuristic and computational approaches for problems that defy classic theoretical solutions. By disproving the existing conjecture, it prompts a reexamination of the theoretical foundations surrounding superpermutations and suggests that similar methodologies might resolve other mathematical conjectures.

Theoretically, the findings challenge existing assumptions within combinatorial optimization and permutation theory, urging further exploration of computationally driven methods and their potential to uncover new mathematical insights. Future developments may include refining heuristic methods for solving larger instances or even achieving exact solutions with more sophisticated algorithms or increased computational resources.

In summary, Houston's investigation into minimal superpermutations offers compelling evidence of both the necessity of rethinking established conjectures and recognizing the role of computational heuristic solvers in advancing mathematical understanding. The results pave the way for further inquiry into the intersection of computation and combinatorial mathematics.