eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-07-04
72:1
72:14
10.4230/LIPIcs.ICALP.2019.72
article
Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits
Hrubeš, Pavel
1
Natarajan Ramamoorthy, Sivaramakrishnan
2
Rao, Anup
2
Yehudayoff, Amir
3
Institute of Mathematics of ASCR, Prague
Paul G. Allen School of Computer Science & Engineering, University of Washington, USA
Department of Mathematics, Technion-IIT, Haifa, Israel
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 Hegedűs 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol132-icalp2019/LIPIcs.ICALP.2019.72/LIPIcs.ICALP.2019.72.pdf
Balancing sets
depth-2 threshold circuits
polynomials
majority
weighted thresholds