All Permutations Supersequence is coNP-complete

TL;DR

This paper proves that determining whether a word contains all permutations as subsequences is coNP-complete, even under certain guarantees, and also establishes NP-completeness for a related graph matching problem.

Contribution

It introduces the coNP-completeness of the all permutations supersequence problem and the NP-completeness of a related non-crossing matching problem.

Findings

Deciding all permutations as subsequences is coNP-complete.

The problem remains hard even with subsequence guarantees.

Related non-crossing matching problem is NP-complete.

Abstract

We prove that deciding whether a given input word contains as subsequence every possible permutation of integers $\{1,2,\ldots,n\}$ is coNP-complete. The coNP-completeness holds even when given the guarantee that the input word contains as subsequences all of length $n-1$ sequences over the same set of integers. We also show NP-completeness of a related problem of Partially Non-crossing Perfect Matching in Bipartite Graphs, i.e. to find a perfect matching in an ordered bipartite graph where edges of the matching incident to selected vertices (even only from one side) are non-crossing.