Unshuffling a Square is NP-Hard

TL;DR

This paper proves that determining whether a string is a square (a shuffle of two identical strings) is NP-hard, resolving an open question by establishing its NP-completeness through a reduction from 3-Partition.

Contribution

It introduces the first proof that recognizing square strings is NP-hard, filling a gap in understanding the computational complexity of string shuffling problems.

Findings

Proves the NP-hardness of recognizing square strings.

Establishes the problem's NP-completeness via reduction from 3-Partition.

Addresses an open question in string shuffle complexity.

Abstract

A shuffle of two strings is formed by interleaving the characters into a new string, keeping the characters of each string in order. A string is a square if it is a shuffle of two identical strings. There is a known polynomial time dynamic programming algorithm to determine if a given string z is the shuffle of two given strings x,y; however, it has been an open question whether there is a polynomial time algorithm to determine if a given string z is a square. We resolve this by proving that this problem is NP-complete via a many-one reduction from 3- Partition.