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Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

Authors Noga Alon, Mrinal Kumar, Ben Lee Volk

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic Complexity
  • Multilinear Circuits
  • Circuit Lower Bounds

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Abstract

We prove a lower bound of Omega(n^2/log^2 n) on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial f(x_1, ..., x_n). Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff ([Ran Raz et al., 2008]), who proved a lower bound of Omega(n^{4/3}/log^2 n) for the same polynomial. Our improvement follows from an asymptotically optimal lower bound for a generalized version of Galvin's problem in extremal set theory.

Cite As Get BibTex

Noga Alon, Mrinal Kumar, and Ben Lee Volk. Unbalancing Sets and an Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CCC.2018.11

Author Details

Noga Alon
  • Sackler School of Mathematics and Blavatnik School of Computer Science , Tel Aviv, 6997801, Israel, and , Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
Mrinal Kumar
  • Center for Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA
Ben Lee Volk
  • Blavatnik School of Computer Science, Tel Aviv University , Tel Aviv, 6997801, Israel

Funding

  • Alon, Noga: Research supported in part by an ISF grant and by a GIF grant.
  • Kumar, Mrinal: Part of this work was done while visiting Tel Aviv University.
  • Volk, Ben Lee: The research leading to these results has received funding from the Israel Science Foundation (grant number 552/16).

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