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Counting magic squares in quasi-polynomial time

Alexander Barvinok·Alex Samorodnitsky·Alexander Yong
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Abstract

We present a randomized algorithm, which, given positive integers n and t and a real number 0< epsilon <1, computes the number Sigma(n, t) of n x n non-negative integer matrices (magic squares) with the row and column sums equal to t within relative error epsilon. The computational complexity of the algorithm is polynomial in 1/epsilon and quasi-polynomial in N=nt, that is, of the order N^{log N}. A simplified version of the algorithm works in time polynomial in 1/epsilon and N and estimates Sigma(n,t) within a factor of N^{log N}. This simplified version has been implemented. We present results of the implementation, state some conjectures, and discuss possible generalizations.

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Alexander Barvinok, Alex Samorodnitsky, Alexander Yong. 2007-03-08. Counting magic squares in quasi-polynomial time. https://arxiv.org/abs/math/0703227

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