Polynomial-time sortable stacks of burnt pancakes

TL;DR

This paper advances understanding of burnt pancake sorting by establishing a new lower bound and demonstrating that simple permutations can be sorted optimally in polynomial time.

Contribution

It introduces a new lower bound for sorting burnt pancakes and shows that simple permutations are sortable in polynomial time, a significant step forward.

Findings

Established a new lower bound for burnt pancake sorting.

Proved simple permutations can be sorted optimally in polynomial time.

Contributed to the complexity understanding of burnt pancake problems.

Abstract

Pancake flipping, a famous open problem in computer science, can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible. In that context, a prefix reversal of length k reverses the order of the first k elements of the permutation. The burnt variant of pancake flipping involves permutations of signed integers, and reversals in that case not only reverse the order of elements but also invert their signs. Although three decades have now passed since the first works on these problems, neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known. In this work, we prove a new lower bound for sorting burnt pancakes, and show that an important class of permutations, known as "simple permutations", can be optimally sorted in polynomial time.