
The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches
We resolve the computational complexity of two problems known as NECKLAC...
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Interpolation of scattered data in R^3 using minimum L_pnorm networks, 1<p<∞
We consider the extremal problem of interpolation of scattered data in R...
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The Classes PPAk: Existence from Arguments Modulo k
The complexity classes PPAk, k ≥ 2, have recently emerged as the main c...
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The Hairy Ball Problem is PPADComplete
The Hairy Ball Theorem states that every continuous tangent vector field...
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Extending partial isometries of antipodal graphs
We prove EPPA (extension property for partial automorphisms) for all ant...
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On the Complexity of Moduloq Arguments and the ChevalleyWarning Theorem
We study the search problem class PPA_q defined as a moduloq analog of ...
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New relations and separations of conjectures about incompleteness in the fnite domain
Our main results are in the following three sections: 1. We prove new ...
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A Topological Characterization of Modulop Arguments and Implications for Necklace Splitting
The classes PPAp have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with p thieves, for prime p. However, these classes are not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the problem. On the contrary, such problems have been pivotal in obtaining completeness results for PPAD and PPA, for several important problems, such as finding a Nash equilibrium [Daskalakis et al., 2009, Chen et al., 2009b] and Necklace Splitting with 2 thieves [FilosRatsikas and Goldberg, 2019]. In this paper, we provide the first topological characterization of the classes PPAp. First, we show that the computational problem associated with a simple generalization of Tucker's Lemma, termed ppolygonTucker, as well as the associated BorsukUlamtype theorem, ppolygonBorsukUlam, are PPApcomplete. Then, we show that the computational version of the wellknown BSS Theorem [Barany et al., 1981], as well as the associated BSSTucker problem are PPApcomplete. Finally, using a different generalization of Tucker's Lemma (termed ℤ_pstarTucker), which we prove to be PPApcomplete, we prove that pthieves Necklace Splitting is in PPAp. All of our containment results are obtained through a new combinatorial proof for ℤ_pversions of Tucker's lemma that is a natural generalization of the standard combinatorial proof of Tucker's lemma by Freund and Todd [1981]. We believe that this new proof technique is of independent interest.
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