Singularity of Sparse Circulant Matrices is NP-complete

Ilia Toli

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arXiv · 0909.2694

Singularity of Sparse Circulant Matrices is NP-complete

Ilia Toli

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Abstract

It is shown by Karp reduction that deciding the singularity of $(2^n - 1) \times (2^n - 1)$ sparse circulant matrices (SC problem) is NP-complete. We can write them only implicitly, by indicating values of the $2 + n(n + 1)/2$ eventually nonzero entries of the first row and can make all matrix operations with them. The positions are $0, 1, 2^{i} + 2^{j}$. The complexity parameter is $n$. Mulmuley's work on the rank of matrices \cite{Mulmuley87} makes SC stand alone in a list of 3,000 and growing NP-complete problems.

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Ilia Toli. 2009-09-14. Singularity of Sparse Circulant Matrices is NP-complete. https://arxiv.org/abs/0909.2694

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