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Revision #1 to TR10-188 | 21st May 2011 21:35

Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae

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Abstract:

We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Our algorithm runs in time $s^{O(1)}\cdot n^{k^{O(k)}}$, where $s$ denotes the size of the formula, $n$ denotes the number of variables, and $k$ bounds the number of occurrences of each variable. Before our work no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time $n^{k^{O(k)} + O(k \log n)}$ in general, and time $n^{k^{O(k^2)} + O(kD)}$ for depth $D$. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae, and for multilinear depth-four formulae.



Changes to previous version:

Extension from multilinear to structurally-multilinear formulae.
Alternate structural witness argument that avoids the use of the rank bound, making the paper self-contained.


Paper:

TR10-188 | 8th December 2010 21:07

Derandomizing Polynomial Identity Testing for Multilinear Constant-Read Formulae


Abstract:

We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Our algorithm runs in time $s^{O(1)}\cdot n^{k^{O(k)}}$, where $s$ denotes the size of the formula, $n$ denotes the number of variables, and $k$ bounds the number of occurrences of each variable. Before our work no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time $n^{k^{O(k)} + O(k \log n)}$ in general, and time $n^{k^{O(k^2)} + O(kd)}$ for depth $d$. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae, and for multilinear depth-four formulae.



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