Black-box identity testing of depth-4 multilinear circuits

Shubhangi Saraf, Ilya Volkovich

Research output: Chapter in Book/Report/Conference proceedingConference contribution

35 Scopus citations


We study the problem of identity testing for multilinear ΣΠΣΠ(k) circuits, i.e. multilinear depth-4 circuits with fan-in k at the top + gate. We give the first polynomial-time deterministic identity testing algorithm for such circuits. Our results also hold in the black-box setting. The running time of our algorithm is (ns)O(k3), where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. The importance of this model arises from [3], where it was shown that derandomizing black-box polynomial identity testing for general depth-4 circuits implies a derandomization of polynomial identity testing (PIT) for general arithmetic circuits. Prior to our work, the best PIT algorithm for multilinear ΣΠΣΠ(k) circuits [13] ran in quasi-polynomial- time, with the running time being nO(k6 log(k) log2 s ). We obtain our results by showing a strong structural result for multilinear ΣΠΣΠ(k) circuits that compute the zero polynomial. We show that under some mild technical conditions, any gate of such a circuit must compute a sparse polynomial. We then show how to combine the structure theorem with a result by Klivans and Spielman [17], on the identity testing for sparse polynomials, to yield the full result.

Original languageEnglish (US)
Title of host publicationSTOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing
Number of pages10
StatePublished - 2011
Externally publishedYes
Event43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States
Duration: Jun 6 2011Jun 8 2011

Publication series

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


Other43rd ACM Symposium on Theory of Computing, STOC'11
Country/TerritoryUnited States
CitySan Jose, CA

All Science Journal Classification (ASJC) codes

  • Software


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


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