Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in

Zohar S. Karnin, Partha Mukhopadhyay, Amir Shpilka, Ilya Volkovich

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We give the first sub-exponential time deterministic polynomial identity testing algorithm for depth-4 multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-4 circuits with a plus gate at the top (also known as ΣΠΣΠ circuits) and has a running time of exp(poly(log(n),log(s),k)) where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. In particular, when the circuit is of polynomial (or quasi-polynomial) size, our algorithm runs in quasi-polynomial time. In [AV08], it was shown that derandomizing polynomial identity testing for general ΣΠΣΠ circuits implies a derandomization of polynomial identity testing in general arithmetic circuits. Prior to this work sub-exponential time deterministic algorithms were known for depth-3 circuits with small top fan-in and for very restricted versions of depth-4 circuits. The main ingredient in our proof is a new structural theorem for multilinear ΣΠΣΠ(k) circuits. Roughly, this theorem shows that any nonzero multilinear ΣΠΣΠ(k) circuit contains an 'embedded' nonzero multilinear ΣΠΣ(k) circuit. Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for multilinear ΣΠΣΠ(k) circuits.

Original languageEnglish
Title of host publicationSTOC'10 - Proceedings of the 2010 ACM International Symposium on Theory of Computing
Number of pages9
StatePublished - 2010
Externally publishedYes
Event42nd ACM Symposium on Theory of Computing, STOC 2010 - Cambridge, MA, United States
Duration: 5 Jun 20108 Jun 2010

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference42nd ACM Symposium on Theory of Computing, STOC 2010
Country/TerritoryUnited States
CityCambridge, MA


  • arithmetic circuits
  • bounded depth circuits
  • derandomization
  • identity testing
  • multilinear circuits


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