  Let $G=(V,E)$ be an undirected graph, $L_G\in \mathbb{R}^{V \times V}$ be the
associated Laplacian matrix, and $b \in \mathbb{R}^V$ be a vector. Solving the
Laplacian system $L_G x = b$ has numerous applications in theoretical computer
science, machine learning, and network analysis. Recently, the notion of the
Laplacian operator $L_F:\mathbb{R}^V \to 2^{\mathbb{R}^V}$ for a submodular
transformation $F:2^V \to \mathbb{R}_+^E$ was introduced, which can handle
undirected graphs, directed graphs, hypergraphs, and joint distributions in a
unified manner. In this study, we show that the submodular Laplacian system
$L_F( x) \ni b$ can be solved in polynomial time. Furthermore, we also prove
that even when the submodular Laplacian system has no solution, we can solve
its regression form in polynomial time. Finally, we discuss potential
applications of submodular Laplacian systems in machine learning and network
analysis.
