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The Multiple-Orientability Thresholds for Random Hypergraphs†

Part of: Algorithms - Computer Science Theory of computing Graph theory

Published online by Cambridge University Press:  28 December 2015

NIKOLAOS FOUNTOULAKIS ,
MEGHA KHOSLA  and
KONSTANTINOS PANAGIOTOU
NIKOLAOS FOUNTOULAKIS
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, B15 2TT, UK (e-mail: fountoun@bham.ac.uk)
MEGHA KHOSLA
Affiliation:
Max Planck Institute for Informatics, Campus E1 4, 66123 Saarbrücken, Germany (e-mail: mkhosla@mpi-inf.mpg.de)
KONSTANTINOS PANAGIOTOU
Affiliation:
Mathematics Institute, University of Munich, Theresienstr. 39, 80333 München, Germany (e-mail: kpanagio@math.lmu.de)
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Abstract

A k-uniform hypergraph H = (V, E) is called ℓ-orientable if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than ℓ edges. Let Hn,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the ℓ-orientability of Hn,m,k for all k ⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantity c*k,ℓ such that with probability 1 − o(1) the graph Hn,cn,k has an ℓ-orientation if c < c*k,ℓ , but fails to do so if c > c*k,ℓ .

Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

MSC classification

Primary: 68Q25: Analysis of algorithms and problem complexity
Secondary: 05C80: Random graphs 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) 68W20: Randomized algorithms 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 6 , November 2016 , pp. 870 - 908
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

†

An extended abstract of this work appeared in the Proceedings of the 22nd ACM–SIAM Symposium on Discrete Algorithms: SODA'11.

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