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

TL;DR

This paper explores the structural properties and relationships of the complexity classes PPA-$k$, which are relevant for problems like fair division and necklace splitting, establishing foundational results for their further study.

Contribution

It provides new structural insights, equivalent definitions, and relationships of PPA-$k$ classes, enhancing understanding of their role in computational complexity.

Findings

PPA-$k$ classes have equivalent definitions.

Relationships between PPA-$k$ and other TFNP classes are clarified.

PPA-$k$ classes are shown to be closed under Turing reductions.

Abstract

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.