THE PHASE TRANSITION OF DISCREPANCY IN RANDOM HYPERGRAPHS
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F23%3A10473965" target="_blank" >RIV/00216208:11320/23:10473965 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OqAHyfWnwN" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=OqAHyfWnwN</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/21M1451427" target="_blank" >10.1137/21M1451427</a>
Alternative languages
Result language
angličtina
Original language name
THE PHASE TRANSITION OF DISCREPANCY IN RANDOM HYPERGRAPHS
Original language description
Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on n vertices and m edges. In the first model, each of the m edges is constructed by placing each vertex into the edge independently with probability d/m, where d is a parameter satisfying d -> infinity and dn/m -> infinity. In the second model, each vertex independently chooses a subset of d edge labels from [m] uniformly at random. Edge i is then defined to be exactly those vertices whose d-subsets include label i. In the sparse regime, i.e., when m = O(n), we show that with high probability a random hypergraph from either model has discrepancy at least Omega(2(-n/m) root dn/m). In the dense regime, i.e., when m >> n, we show that with high probability a random hypergraph from either model has discrepancy at least Omega(root(dn/m) log gamma), where gamma = min{m/n, dn/m}. Furthermore, we obtain nearly matching asymptotic upper bounds on the discrepancy. Specifically, we apply the partial coloring lemma of Lovett and Meka to show that, in the dense regime, with high probability the two random hypergraph models each have discrepancy O(root dn/m log(m/n)). In fact, in a significant parameter range we can tighten our analysis to get an upper bound which matches our lower bound up to a constant factor. This result is algorithmic, and together with the work of Bansal and Meka [On the discrepancy of random low degree set systems, in Proceedings of the 2019 Annual ACM-SIAM Symposium on Discrete Algorithms, 2019, pp. 2557-2564] characterizes how the discrepancy of each random hypergraph transitions from Theta (surd d) to o(surd d) as m increases from m= Theta (n) to m >> n.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Discrete Mathematics
ISSN
0895-4801
e-ISSN
1095-7146
Volume of the periodical
37
Issue of the periodical within the volume
3
Country of publishing house
US - UNITED STATES
Number of pages
24
Pages from-to
1818-1841
UT code for WoS article
001071676800016
EID of the result in the Scopus database
2-s2.0-85171614345