Tight bounds on the maximum size of a set of permutations with bounded VC-dimension

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F12%3A10125751" target="_blank" >RIV/00216208:11320/12:10125751 - isvavai.cz</a>
Result on the web
<a href="http://siam.omnibooksonline.com/2012SODA/data/papers/243.pdf" target="_blank" >http://siam.omnibooksonline.com/2012SODA/data/papers/243.pdf</a>
DOI - Digital Object Identifier
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Result language
angličtina
Original language name
Tight bounds on the maximum size of a set of permutations with bounded VC-dimension
Original language description
The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let r_k(n) be the maximum size of a set of n-permutations with VC-dimension k. Raz showed that r_2(n) grows exponentially in n. We show that r_3(n)=2^Theta(n log(alpha(n))) and for every s }= 4, we have almost tight upper and lower bounds of the form 2^{n poly(alpha(n))}. We also study the maximum number p_k(n) of 1-entries in an n x n (0,1)-matrix with no (k+1)-tuple of columns containing all (k+1)-permutation matrices. We determine that p_3(n) = Theta(n alpha(n)) and that p_s(n) can be bounded by functions of the form n 2^poly(alpha(n)) for every fixed s }= 4. We also show that for every positive s there is a slowly growing function zeta_s(m) (of the form 2^poly(alpha(m)) for every fixed s }= 5) satisfying the following. For all positive integers n and B and every n x n (0,1)-matrix
Czech name
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Czech description
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Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Publication year
2012
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
ISBN
978-1-61197-211-5
ISSN
2160-1445
e-ISSN
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Number of pages
10
Pages from-to
1113-1122
Publisher name
SIAM
Place of publication
PHILADELPHIA
Event location
Japonsko
Event date
Jan 17, 2012
Type of event by nationality
WRD - Celosvětová akce
UT code for WoS article
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