Pancake Flipping is hard

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 for 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.

[1]  Vineet Bafna,et al.  Genome Rearrangements and Sorting by Reversals , 1996, SIAM J. Comput..

[2]  Ivan Hal Sudborough,et al.  An (18/11)n upper bound for sorting by prefix reversals , 2009, Theor. Comput. Sci..

[3]  D. J. Kleitman,et al.  Elementary Problems: E2564-E2569 , 1975 .

[4]  Sheldon B. Akers,et al.  A Group-Theoretic Model for Symmetric Interconnection Networks , 1989, IEEE Trans. Computers.

[5]  Josef Cibulka,et al.  On average and highest number of flips in pancake sorting , 2009, Theor. Comput. Sci..

[6]  Manuel Blum,et al.  on the Problem of Sorting Burnt Pancakes , 1995, Discret. Appl. Math..

[7]  Marek Karpinski,et al.  1.375-Approximation Algorithm for Sorting by Reversals , 2002, ESA.

[8]  Guillaume Fertin,et al.  Pancake Flipping Is Hard , 2011, MFCS.

[9]  Ivan Hal Sudborough,et al.  On Sorting by Prefix Reversals and the Diameter of Pancake Networks , 1992, Heinz Nixdorf Symposium.

[10]  Selim G. Akl,et al.  Parallel Routing and Sorting of the Pancake Network , 1991, ICCI.

[11]  Pavel A. Pevzner,et al.  Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals , 1999, J. ACM.

[12]  Selim G. Akl,et al.  On Some Properties and Algorithms for the Star and Pancake Interconnection Networks , 1994, J. Parallel Distributed Comput..

[13]  Marek Karpinski,et al.  On Some Tighter Inapproximability Results (Extended Abstract) , 1999, ICALP.

[14]  Ivan Hal Sudborough,et al.  On the Diameter of the Pancake Network , 1997, J. Algorithms.

[15]  Johannes Fischer,et al.  A 2-Approximation Algorithm for Sorting by Prefix Reversals , 2005, ESA.

[16]  Josef Cibulka,et al.  Polynomial-time sortable stacks of burnt pancakes , 2011, Theor. Comput. Sci..

[17]  Christos H. Papadimitriou,et al.  Bounds for sorting by prefix reversal , 1979, Discret. Math..