Unit-Weight Laplacians are Complete for Linear Systems Modulo p
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In this paper, we prove that over finite fields modulo primes, solving general linear systems is as hard as solving unit-weight Laplacian linear systems. We give a reduction of solving a general linear system 𝐀x = b over ℤ_p to solving a unit-weight Laplacian system 𝐋̅ of size O(nnz(𝐀)log^2p/loglog p). Our result indicates that unlike problems over reals, graph-like structure such as Laplacians may not offer too many additional properties over finite fields. We also formalize the role of Schur complement as a tool for making reductions between problems on systems of linear equations.
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