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Improved Bounds for Randomly Colouring Simple Hypergraphs

Authors Weiming Feng , Heng Guo , Jiaheng Wang



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Weiming Feng
  • School of Informatics, University of Edinburgh, UK
Heng Guo
  • School of Informatics, University of Edinburgh, UK
Jiaheng Wang
  • School of Informatics, University of Edinburgh, UK

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Weiming Feng, Heng Guo, and Jiaheng Wang. Improved Bounds for Randomly Colouring Simple Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 25:1-25:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.25

Abstract

We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree Δ. For any δ > 0, if k ≥ 20(1+δ)/δ and q ≥ 100Δ^({2+δ}/{k-4/δ-4}), the running time of our algorithm is Õ(poly(Δ k)⋅ n^1.01), where n is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021), and does not require Ω(log n) colours unlike the work of Frieze and Anastos (2017).

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Approximate counting
  • Markov chain
  • Mixing time
  • Hypergraph colouring

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