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The Composition Complexity of Majority

Authors Victor Lecomte, Prasanna Ramakrishnan, Li-Yang Tan

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  • Theory of computation → Circuit complexity
Keywords
  • computational complexity
  • circuit lower bounds

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Abstract

We study the complexity of computing majority as a composition of local functions: Maj_n = h(g_1,…,g_m), where each g_j: {0,1}ⁿ → {0,1} is an arbitrary function that queries only k ≪ n variables and h: {0,1}^m → {0,1} is an arbitrary combining function. We prove an optimal lower bound of m ≥ Ω(n/k log k) on the number of functions needed, which is a factor Ω(log k) larger than the ideal m = n/k. We call this factor the composition overhead; previously, no superconstant lower bounds on it were known for majority.
Our lower bound recovers, as a corollary and via an entirely different proof, the best known lower bound for bounded-width branching programs for majority (Alon and Maass '86, Babai et al. '90). It is also the first step in a plan that we propose for breaking a longstanding barrier in lower bounds for small-depth boolean circuits. 
Novel aspects of our proof include sharp bounds on the information lost as computation flows through the inner functions g_j, and the bootstrapping of lower bounds for a multi-output function (Hamming weight) into lower bounds for a single-output one (majority).

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Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan. The Composition Complexity of Majority. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 19:1-19:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.19

Author Details

Victor Lecomte
  • Stanford University, CA, USA
Prasanna Ramakrishnan
  • Stanford University, CA, USA
Li-Yang Tan
  • Stanford University, CA, USA

Funding

Victor, Pras, and Li-Yang are supported by NSF CAREER Award CCF-1942123. Pras is also supported by Moses Charikar’s Simons Investigator Award.

Acknowledgements

Li-Yang thanks Xi Chen, Rocco Servedio, and Erik Waingarten for numerous discussions about this problem.

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