TY - GEN
T1 - Lower bounds on balancing sets and depth-2 threshold circuits
AU - Hrubeš, Pavel
AU - Ramamoorthy, Sivaramakrishnan Natarajan
AU - Rao, Anup
AU - Yehudayoff, Amir
N1 - Publisher Copyright: © Graham Cormode, Jacques Dark, and Christian Konrad; licensed under Creative Commons License CC-BY
PY - 2019/7/1
Y1 - 2019/7/1
N2 - There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S1,..., Sk ⊂ [n] is balancing if for every subset X ⊂ {1, 2,..., n} of size n/2, there is an i ∈ [k] so that |Si ∩ X| = |Si|/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.
AB - There are various notions of balancing set families that appear in combinatorics and computer science. For example, a family of proper non-empty subsets S1,..., Sk ⊂ [n] is balancing if for every subset X ⊂ {1, 2,..., n} of size n/2, there is an i ∈ [k] so that |Si ∩ X| = |Si|/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.
KW - Balancing sets
KW - Depth-2 threshold circuits
KW - Majority
KW - Polynomials
KW - Weighted thresholds
UR - http://www.scopus.com/inward/record.url?scp=85069222008&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ICALP.2019.72
DO - 10.4230/LIPIcs.ICALP.2019.72
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 1
EP - 14
BT - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
A2 - Baier, Christel
A2 - Chatzigiannakis, Ioannis
A2 - Flocchini, Paola
A2 - Leonardi, Stefano
T2 - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
Y2 - 9 July 2019 through 12 July 2019
ER -