# Polynomial-Time Algorithms for Submodular Laplacian Systems

## TL;DR

This paper introduces polynomial-time algorithms for solving submodular Laplacian systems, extending classical Laplacian solvers to a broader class of submodular transformations with applications in machine learning and network analysis.

## Contribution

It presents the first polynomial-time algorithms for solving submodular Laplacian systems and their regression forms, generalizing traditional Laplacian system solutions.

## Key findings

- Submodular Laplacian systems can be solved in polynomial time.
- Regression problems for these systems are also solvable efficiently.
- Potential applications include advanced machine learning and network analysis techniques.

## Abstract

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.