THE PHASE TRANSITION OF DISCREPANCY IN RANDOM HYPERGRAPHS

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

THE PHASE TRANSITION OF DISCREPANCY IN RANDOM HYPERGRAPHS

Popis výsledku v původním jazyce

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 -&gt; infinity and dn/m -&gt; 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 &gt;&gt; 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 &gt;&gt; n.

Název v anglickém jazyce

THE PHASE TRANSITION OF DISCREPANCY IN RANDOM HYPERGRAPHS

Popis výsledku anglicky

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 -&gt; infinity and dn/m -&gt; 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 &gt;&gt; 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 &gt;&gt; n.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

SIAM Journal on Discrete Mathematics

ISSN

0895-4801

e-ISSN

1095-7146

Svazek periodika

37

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

24

Strana od-do

1818-1841

Kód UT WoS článku

001071676800016

EID výsledku v databázi Scopus

2-s2.0-85171614345