Permutation Reconstruction from Differences

TL;DR

This paper proves that reconstructing a permutation from absolute differences is NP-complete, highlighting its computational intractability, and introduces the NP-completeness of a new puzzle called Crazy Frog Puzzle.

Contribution

It establishes the NP-completeness of permutation reconstruction from differences and introduces Crazy Frog Puzzle as a new NP-complete problem.

Findings

Permutation reconstruction from differences is NP-complete.

Crazy Frog Puzzle is NP-complete.

Permutation problem is among the simplest combinatorial problems proven intractable.

Abstract

We prove that the problem of reconstructing a permutation $\pi_1,\dotsc,\pi_n$ of the integers $[1\dotso n]$ given the absolute differences $|\pi_{i+1}-\pi_i|$, $i = 1,\dotsc,n-1$ is NP-complete. As an intermediate step we first prove the NP-completeness of the decision version of a new puzzle game that we call Crazy Frog Puzzle. The permutation reconstruction from differences is one of the simplest combinatorial problems that have been proved to be computationally intractable.