Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Bläsius, Thomas; Friedrich, Tobias; Schirneck, Martin https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-128871
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The Minimization of Random Hypergraphs

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Abstract

We investigate the maximum-entropy model B_{n,m,p} for random n-vertex, m-edge multi-hypergraphs with expected edge size pn. We show that the expected size of the minimization min(B_{n,m,p}), i.e., the number of inclusion-wise minimal edges of B_{n,m,p}, undergoes a phase transition with respect to m. If m is at most 1/(1-p)^{(1-p)n}, then E[|min(B_{n,m,p})|] is of order Θ(m), while for m ≥ 1/(1-p)^{(1-p+ε)n} for any ε > 0, it is Θ(2^{(H(α) + (1-α) log₂ p) n}/√n). Here, H denotes the binary entropy function and α = - (log_{1-p} m)/n. The result implies that the maximum expected number of minimal edges over all m is Θ((1+p)ⁿ/√n). Our structural findings have algorithmic implications for minimizing an input hypergraph. This has applications in the profiling of relational databases as well as for the Orthogonal Vectors problem studied in fine-grained complexity. We make several technical contributions that are of independent interest in probability. First, we improve the Chernoff-Hoeffding theorem on the tail of the binomial distribution. In detail, we show that for a binomial variable Y ∼ Bin(n,p) and any 0 < x < p, it holds that P[Y ≤ xn] = Θ(2^{-D(x‖p) n}/√n), where D is the binary Kullback-Leibler divergence between Bernoulli distributions. We give explicit upper and lower bounds on the constants hidden in the big-O notation that hold for all n. Secondly, we establish the fact that the probability of a set of cardinality i being minimal after m i.i.d. maximum-entropy trials exhibits a sharp threshold behavior at i^* = n + log_{1-p} m.

BibTeX - Entry

@InProceedings{blsius_et_al:LIPIcs:2020:12887,
  author =	{Thomas Bl{\"a}sius and Tobias Friedrich and Martin Schirneck},
  title =	{{The Minimization of Random Hypergraphs}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12887},
  URN =		{urn:nbn:de:0030-drops-128871},
  doi =		{10.4230/LIPIcs.ESA.2020.21},
  annote =	{Keywords: Chernoff-Hoeffding theorem, maximum entropy, maximization, minimization, phase transition, random hypergraphs}
}

Keywords: Chernoff-Hoeffding theorem, maximum entropy, maximization, minimization, phase transition, random hypergraphs
Seminar: 28th Annual European Symposium on Algorithms (ESA 2020)
Issue date: 2020
Date of publication: 26.08.2020


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