The Classes PPA-k: Existence from Arguments Modulo k

Abstract

The complexity classes PPA-k, k ≥ 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.

A Topological Characterization of Modulo-$p$ Arguments and Implications for Necklace Splitting

“…Computational Classes: As mentioned earlier, among the classes of TFNP, PPAD has been the most successful in capturing the complexity of well-known problems. Besides the complexity of computing a Nash equilibrium [Daskalakis et al, 2009, Chen et al, 2009b, Rubinstein, 2018, other PPAD-complete problems are computing equilibria in markets [Chen et al, 2009a[Chen et al, , 2013, versions of envy-free cake cutting [Deng et al, 2012] and fixed-point theorems [Mehta, 2014, Goldberg andHollender, 2019].…”

“…Recently, Göös et al [2019] showed that an explicit version of the problem is complete for PPA-p, therefore obtaining the first PPA-p-completeness result for a natural problem. The authors of [Göös et al, 2019], as well as Hollender [2019], independently also extended the definition of the classes PPA-k to any k ≥ 2, and provided several characterizations in terms of their defined-for-primes counterparts. Hollender [2019] also investigated the connection with the classes PMOD-k, which bear strong resemblance to PPA-k, and were defined seemingly independently of Papadimitriou's work by Johnson [2011].…”

“…A simple way to visualize this argument is to imagine that if a space has cardinality that is non-zero modulo k and if we can group points in this space that are nonsolutions into groups of size k, then in this space there should exist a solution. This kind of existential argument has been formalized by Papadimitriou in his seminal paper [Papadimitriou, 1994] and there have been various instantiations of this principle from which we can define corresponding computational problems [Göös et al, 2019, Hollender, 2019. In this paper, we mostly rely on the following instantiation defined by Hollender [2019]:…”

“…This kind of existential argument has been formalized by Papadimitriou in his seminal paper [Papadimitriou, 1994] and there have been various instantiations of this principle from which we can define corresponding computational problems [Göös et al, 2019, Hollender, 2019. In this paper, we mostly rely on the following instantiation defined by Hollender [2019]:…”

“…The classes PPA-p have been very recently studied by Hollender [2019] and Göös et al [2019]. The authors of the latter paper in fact provide the first PPA-p-completeness result for a natural problem, a computational version of the Chevalley-Warning theorem.…”