An upper bound for permanents of nonnegative matrices

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Abstract

A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in lp is attained either for the identity matrix I or for a constant multiple of the all-1 matrix J. The conjecture is known to be true for p = 1 (I) and for p ≥ 2 (J). We prove the conjecture for a subinterval of (1, 2), and show the conjectured upper bound to be true within a subexponential factor (in the dimension) for all 1 < p < 2. In fact, for p bounded away from 1, the conjectured upper bound is true within a constant factor. © 2007 Elsevier Inc. All rights reserved.

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Samorodnitsky, A. (2008). An upper bound for permanents of nonnegative matrices. Journal of Combinatorial Theory. Series A, 115(2), 279–292. https://doi.org/10.1016/j.jcta.2007.05.010

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