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Is there any known efficient algorithm (something that works better than brute-force algorithm) to check that given a $(0,1)$-$d \times d$ matrix $A$, is there a permutation matrix $P$ of the same size such that $PA$ is a symmetric matrix? I believe this question might have been done in the past (probably even for a more general matrix), however I can not find any literature that is connected to this problem.

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    $\begingroup$ Bhat, Kabekode VS. "An algorithm for matrix symmetrization." Journal of the Franklin Institute 312.1 (1981): 89-93. $\endgroup$ – Waldemar Jan 23 '14 at 10:04
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The problem is NP-complete, see C. Colbourn and B. D. McKay, A correction to Colbourn's paper on the complexity of matrix symmetrizability, Information Processing Letters, 11 (1980) 96-97. Here is a scan.

Interestingly, if we ask instead whether there is a permutation matrix $P$ such that $PA$ is the transpose of $A$, that is equivalent to the graph isomorphism problem.

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