Convex Submodular Minimization with Indicator Variables

TL;DR

This paper introduces a method to solve convex submodular optimization problems with indicator variables by reducing them to binary submodular minimization, enabling efficient solutions for applications like MRF inference.

Contribution

It shows that convex submodular problems with indicator variables can be reduced to binary submodular minimization and develops a parametric approach for fast solutions.

Findings

Problems are strongly polynomially solvable after reformulation.

A parametric approach efficiently computes extreme bases.

Numerical experiments demonstrate the method's efficiency.

Abstract

We study a general class of convex submodular optimization problems with indicator variables. Many applications such as the problem of inferring Markov random fields (MRFs) with a sparsity or robustness prior can be naturally modeled in this form. We show that these problems can be reduced to binary submodular minimization problems, possibly after a suitable reformulation, and thus are strongly polynomially solvable. Furthermore, we develop a parametric approach for computing the associated extreme bases under certain smoothness conditions. This leads to a fast solution method, whose efficiency is demonstrated through numerical experiments.