Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-bio
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The categories of this paper are: patt-sol
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The categories of this paper are: astro-ph
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The categories of this paper are: math-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin