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On the Central Levels Problem

Authors Petr Gregor, Ondřej Mička, Torsten Mütze

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ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Matchings and factors
Keywords
  • Gray code
  • Hamilton cycle
  • hypercube
  • middle levels
  • symmetric chain decomposition

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Abstract

The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-𝓁 many 1s and at most m+𝓁 many 1s, i.e., the vertices in the middle 2𝓁 levels, has a Hamilton cycle for any m ≥ 1 and 1 ≤ 𝓁 ≤ m+1. This problem was raised independently by Savage, by Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case 𝓁 = 1, and classical binary Gray codes, namely the case 𝓁 = m+1. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of 𝓁 consecutive levels in the n-dimensional hypercube for any n ≥ 1 and 1 ≤ 𝓁 ≤ n+1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n-dimensional hypercube, n≥ 2, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code.

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Petr Gregor, Ondřej Mička, and Torsten Mütze. On the Central Levels Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 60:1-60:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.60

Author Details

Petr Gregor
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic
Ondřej Mička
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic
Torsten Mütze
  • Department of Computer Science, University of Warwick, Coventry, UK
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic

Funding

This work was supported by the Czech Science Foundation grant GA19-08554S.
  • Mütze, Torsten: was also supported by DFG grant 413902284.

Acknowledgements

We thank Jiří Fink for several valuable discussions about symmetric chain decompositions, and for feedback on an earlier draft of this paper.

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