Lower bounds on balancing sets and depth-2 threshold circuits

0507746 - MÚ 2020 RIV DE eng C - Conference Paper (international conference)
Hrubeš, Pavel - Natarajan Ramamoorthy, S. - Rao, A. - Yehudayoff, A.
Lower bounds on balancing sets and depth-2 threshold circuits.
46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Dagstuhl: Schloss Dagstuhl, Leibniz-Zentrum für Informatik, 2019 - (Baier, C.; Chatzigiannakis, I.; Flocchini, P.; Leonardi, S.), č. článku 72. Leibniz International Proceedings in Informatics (LIPIcs), 132. ISBN 978-3-95977-109-2. ISSN 1868-8969.
[46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Patras (GR), 08.07.2019-12.07.2019]
R&D Projects: GA ČR(CZ) GX19-27871X
EU Projects: European Commission(XE) 339691 - FEALORA
Institutional support: RVO:67985840
Keywords : balancing sets * depth-2 threshold circuits * polynomials * majority * weighted thresholds
OECD category: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
http://drops.dagstuhl.de/opus/volltexte/2019/10648/

There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S_1,...,S_k subset [n] is balancing if for every subset X subset {1,2,...,n} of size n/2, there is an i in [k] so that |S_i cap X| = |S_i|/2. We extend and simplify the framework developed by Hegedüs for proving lower bounds on the size of balancing set families. We prove that if n=2p for a prime p, then k >= p. For arbitrary values of n, we show that k >= n/2 - o(n). We then exploit the connection between balancing families and depth-2 threshold circuits. This connection helps resolve a question raised by Kulikov and Podolskii on the fan-in of depth-2 majority circuits computing the majority function on n bits. We show that any depth-2 threshold circuit that computes the majority on n bits has at least one gate with fan-in at least n/2 - o(n). We also prove a sharp lower bound on the fan-in of depth-2 threshold circuits computing a specific weighted threshold function.
Permanent Link: http://hdl.handle.net/11104/0298721