The Classes PPA-$k$: Existence from Arguments Modulo $k$

The complexity classes PPA-$k$, $k \geq 2$, have recently emerged as the main candidates for capturing the complexity of important problems in fair division, in particular Alon's Necklace-Splitting problem with $k$ thieves. Indeed, the problem with two thieves has been shown complete for PPA = PPA-2. In this work, we present structural results which provide a solid foundation for the further study of these classes. Namely, we investigate the classes PPA-$k$ in terms of (i) equivalent definitions, (ii) inner structure, (iii) relationship to each other and to other TFNP classes, and (iv) closure under Turing reductions.