On monotone circuits with local oracles and clique lower bounds

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10401463" target="_blank" >RIV/00216208:11320/19:10401463 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=0PU5oLJMKa" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=0PU5oLJMKa</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.4086/cjtcs.2018.001" target="_blank" >10.4086/cjtcs.2018.001</a>

Result language
angličtina
Original language name
On monotone circuits with local oracles and clique lower bounds
Original language description
We investigate monotone circuits with local oracles [K., 2016], i.e., circuits containing additional inputs y i =y i (x ⃗ ) that can perform unstructured computations on the input string x ⃗ . Let μELEMENT OF[0,1] be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions y i (x ⃗ ) , and U n,k ,V n,k SUBSET OF OR EQUAL TO {0,1} m be the set of k -cliques and the set of complete (k-1) -partite graphs, respectively (similarly to [Razborov, 1985]). Our results can be informally stated as follows. 1. For an appropriate extension of depth-2 monotone circuits with local oracles, we show that the size of the smallest circuits separating U n,3 (triangles) and V n,3 (complete bipartite graphs) undergoes two phase transitions according to μ . 2. For 5<=k(n)<=n 1/4 , arbitrary depth, and μ<=1/50 , we prove that the monotone circuit size complexity of separating the sets U n,k and V n,k is n Θ(k SQUARE ROOT ) , under a certain restrictive assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of k -clique obtained by Alon and Boppana (1987).
Czech name
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Czech description
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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