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A natural barrier in random greedy hypergraph matching

Part of: Combinatorial probability Theory of computing Extremal combinatorics

Published online by Cambridge University Press:  27 June 2019

Patrick Bennett  and
Tom Bohman
Patrick Bennett
Affiliation:
Mathematics Department, Western Michigan University, Kalamazoo, MI 49008, USA, Email: patrick.bennett@wmich.edu
Tom Bohman*
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA
*
*Corresponding author. Email: tbohman@math.cmu.edu
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Abstract

Let r ⩾ 2 be a fixed constant and let $ {\cal H} $ be an r-uniform, D-regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that degrees of pairs of vertices in $ {\cal H} $ are at most L where L = D/( log N)ω(1). We consider the random greedy algorithm for forming a matching in $ {\cal H} $. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\cal H} $ that are not saturated by the final matching is at most (L/D)(1/(2(r−1)))+o(1). This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

MSC classification

Primary: 05D40: Probabilistic methods
Secondary: 60C05: Combinatorial probability 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.)
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 6 , November 2019 , pp. 816 - 825
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Research supported in part by NSF grant DMS-1001638 and Simons Foundation grant #426894.

‡

Research supported in part by NSF grants DMS-1001638 and DMS-1100215.

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