Filos-Ratsikas, A 
ORCID: 0000-0001-7868-8114, Hollender, A, Sotiraki, K and Zampetakis, M
(2021)
A topological characterization of modulo-p arguments and implications for necklace splitting.
In: SIAM Symposium on Discrete Algorithms, 2021-1-10 - 2021-1-11, Virtual Conference.
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Abstract
The classes PPA-p 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 were not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the Necklace Splitting problem. On the contrary, topological problems have been pivotal in obtaining completeness results for PPAD and PPA, such as the PPAD-completeness of finding a Nash equilibrium [18, 15] and the PPA-completeness of Necklace Splitting with 2 thieves [24]. In this paper, we provide the first topological characterization of the classes PPA-p. First, we show that the computational problem associated with a simple generalization of Tucker's Lemma, termed p-polygon-Tucker, as well as the associated Borsuk-Ulam-type theorem, p-polygon-Borsuk-Ulam, are PPA-p-complete. Then, we show that the computational version of the well-known BSS Theorem [8], as well as the associated BSS-Tucker problem are PPA-p-complete. Finally, using a different generalization of Tucker's Lemma (termed Zp-star-Tucker), which we prove to be PPA-p-complete, we prove that p-thief Necklace Splitting is in PPA-p. This latter result gives a new combinatorial proof for the Necklace Splitting theorem, the only proof of this nature other than that of Meunier [42]. All of our containment results are obtained through a new combinatorial proof for Zp-versions of Tucker's lemma that is a natural generalization of the standard combinatorial proof of Tucker's lemma by Freund and Todd [27]. We believe that this new proof technique is of independent interest.
| Item Type: | Conference or Workshop Item (Unspecified) |
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| Divisions: | Faculty of Science and Engineering > School of Electrical Engineering, Electronics and Computer Science |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 24 Jun 2021 10:27 |
| Last Modified: | 18 Jan 2023 23:04 |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3112813 |
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