Pancake Flipping is Hard

TL;DR

This paper proves that the classic Pancake Flipping problem is NP-hard, resolving a long-standing open question about its computational complexity and highlighting its significance in theoretical computer science.

Contribution

The paper establishes that the original Pancake Flipping problem is NP-hard, providing a definitive complexity classification for this well-studied problem.

Findings

Proves NP-hardness of the Pancake Flipping problem

Addresses a 30-year open question in computational complexity

Highlights implications for related fields like biology and parallel computing

Abstract

Pancake Flipping is the problem of sorting a stack of pancakes of different sizes (that is, a permutation), when the only allowed operation is to insert a spatula anywhere in the stack and to flip the pancakes above it (that is, to perform a prefix reversal). In the burnt variant, one side of each pancake is marked as burnt, and it is required to finish with all pancakes having the burnt side down. Computing the optimal scenario for any stack of pancakes and determining the worst-case stack for any stack size have been challenges over more than three decades. Beyond being an intriguing combinatorial problem in itself, it also yields applications, e.g. in parallel computing and computational biology. In this paper, we show that the Pancake Flipping problem, in its original (unburnt) variant, is NP-hard, thus answering the long-standing question of its computational complexity.