Abstract
We show that computation of the SPERNER problem is PPA-complete on the Möbius band under proper boundary conditions, settling a long term open problem. Further, the same computational complexity results extend to other discrete fixed points on the Möbius band, such as the Brouwer fixed point problem, the DPZP fixed point problem and a simple version of the Tucker problem, as well as the projective plane and the Klein bottle. We expect it opens up a new route for further studies on the related combinatorial structures.
Highlights
In his seminal work on understanding the time complexity of the parity argument, Papadimitriou introduced the well known class PPAD [27] that has influenced a generation of algorithmic game theorists in their study of economic computations
The class PPA is a superset of PPAD, and the intuitive reason is that directions are helpful: Finding another node of the appropriate kind is harder to solve when there are no directions; oracle separation is known [3]
The class PPAD has many problems that have been shown complete for PPAD such as in the incomplete list of 25 of them [22] gathered by Kintali
Summary
In his seminal work on understanding the time complexity of the parity argument, Papadimitriou introduced the well known class PPAD [27] that has influenced a generation of algorithmic game theorists in their study of economic computations. On the higher constant dimension non-orientable space, all the discrete fixed point problems follow from the 2D results to become PPA-complete. As we prove related results for other non-orientable spaces such as the Klein bottle or projective space, we would have to refer to less natural 3D (in 5 dimensions) Klein solid bottle or 3D projective space, with unbearable complications in the proofs One such case is in the beautiful PSPACE proof of the other end of the line for the path following algorithm in the 2D discrete fixed point proof by Goldberg[16]. Because of the space limitation, we put most of our proofs in the Appendix
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