Black-Box Identity Testing of Depth-4 Multilinear Circuits

Abstract

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 when k=O(1). Our results also hold in the black-box setting.

The running time of our algorithm is \({\left( {ns} \right)^{{\text{O}}\left( {{k^3}} \right)}}\), 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 [11], 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 [31] ran in quasi-polynomial-time, with the running time being \({n^{{\rm O}\left( {{k^6}\log \left( k \right){{\log }^2}s} \right)}}\).

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 [33], on the identity testing for sparse polynomials, to yield the full result.

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Correspondence to Ilya Volkovich.

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Research was partially supported by the European Commission’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 257575.

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Saraf, S., Volkovich, I. Black-Box Identity Testing of Depth-4 Multilinear Circuits. Combinatorica 38, 1205–1238 (2018). https://doi.org/10.1007/s00493-016-3460-4

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Mathematics Subject Classification (2000)

  • 68Q25
  • 12Y05