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2010
ACM

On the Hardness of the Noncommutative Determinant

14 years 8 months ago
On the Hardness of the Noncommutative Determinant
In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below: ? We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent. Consequently, the commutative permanent polynomial has small commutative arithmetic circuits. ? For any field F we show that computing the n ? n permanent over F is polynomial-time reducible to computing the 2n ? 2n (noncommutative) determinant whose entries are O(n2 ) ? O(n2 ) matrices over the field F. ? We also derive as a consequence that computing the n ? n permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant o...
Vikraman Arvind and Srikanth Srinivasan
Added 01 Mar 2010
Updated 02 Mar 2010
Type Conference
Year 2010
Where STOC
Authors Vikraman Arvind and Srikanth Srinivasan
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