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Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits

Authors Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, Amir Yehudayoff

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Pavel Hrubeš
  • Institute of Mathematics of ASCR, Prague
Sivaramakrishnan Natarajan Ramamoorthy
  • Paul G. Allen School of Computer Science & Engineering, University of Washington, USA
Anup Rao
  • Paul G. Allen School of Computer Science & Engineering, University of Washington, USA
Amir Yehudayoff
  • Department of Mathematics, Technion-IIT, Haifa, Israel

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Pavel Hrubeš, Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao, and Amir Yehudayoff. Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 72:1-72:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Theory of computation → Circuit complexity
  • Balancing sets
  • depth-2 threshold circuits
  • polynomials
  • majority
  • weighted thresholds


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