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Permutation Codes, Hamming Graphs and Turán Graphs

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Applied Physics, System Science and Computers III (APSAC 2018)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 574 ))

Abstract

This paper investigates the properties of permutation Hamming graphs, a class of graphs in which the vertices are the permutations of n symbols and the edges connect pairs of vertices at a Hamming distance greater than or equal to a value d. Despite a remarkable regularity, permutation Hamming graphs elude general formulas for relevant indicators like the clique number. The clique number plays a crucial role in the Maximum Permutation Code Problem (MPCP), a well-known optimization problem. This work focuses on the relationship between permutation Hamming graphs and a particular type of Turán graphs. The main result is a theorem asserting that permutation Hamming graphs are the intersection of a set of Turán graphs. This equivalence has implications on the MPCP. In fact it enables a reformulation as a hitting set problem, which in turn can be translated into a binary integer program.

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Author information

Authors and Affiliations

  1. Dalle Molle Institute for Artificial Intelligence IDSIA - USI/SUPSI, 6928, Manno, Switzerland

    János Barta & Roberto Montemanni

Authors
  1. János Barta
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  2. Roberto Montemanni
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Corresponding author

Correspondence to János Barta .

Editor information

Editors and Affiliations

  1. Department of Marketing, University of Applied Sciences, Athens, Greece

    Klimis Ntalianis

  2. School of Electrical and Computer Engineering, The Georgia Institute of Technology, Atlanta, GA, USA

    George Vachtsevanos

  3. Ecole Central de Lille, Lille, France

    Pierre Borne

  4. Faculty of Mathematics, Alexandru Ioan Cuza University, Iasi, Romania

    Anca Croitoru

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Barta, J., Montemanni, R. (2019). Permutation Codes, Hamming Graphs and Turán Graphs. In: Ntalianis, K., Vachtsevanos, G., Borne, P., Croitoru, A. (eds) Applied Physics, System Science and Computers III. APSAC 2018. Lecture Notes in Electrical Engineering, vol 574 . Springer, Cham. https://doi.org/10.1007/978-3-030-21507-1_17

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