Singularity of Sparse Circulant Matrices is NP-complete

TL;DR

This paper proves that determining the singularity of large sparse circulant matrices, which can be represented implicitly, is an NP-complete problem, highlighting its computational difficulty.

Contribution

It establishes the NP-completeness of the singularity decision problem for sparse circulant matrices, a significant addition to known NP-complete problems.

Findings

Deciding singularity is NP-complete for these matrices.

Matrices can be represented implicitly via specific nonzero entries.

The problem's complexity parameter is n.

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.