Unit-Weight Laplacians are Complete for Linear Systems Modulo $p$

In this paper, we prove that over the finite field $\mathbb{Z}_p$, solving unit-weight Laplacian linear systems is as hard as solving general linear systems. The proof includes the following two reductions. (1) Reduce solving a general linear system $\mathbf{A}$ to solving a weighted Laplacian system $\mathbf{L}$ of size $O(\mathrm{nnz}(\mathbf{A}))$. (2) Reduce solving a weighted Laplacian system $\mathbf{L}$ to solving a unit-weight Laplacian system $\bar{\mathbf{L}}$ of size $O\left(\mathrm{nnz}(\mathbf{L})\log^2p/\log\log p\right)$.