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 with practical numerical demonstrations.

Contribution

It establishes a reduction technique for a broad class of convex submodular problems to binary submodular minimization, and develops a parametric approach for computing extreme bases efficiently.

Findings

Problems are strongly polynomially solvable after reformulation.

A fast solution method is demonstrated through numerical experiments.

Implications for quadratic objectives are discussed.

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. %We also discuss the implication of our results in the case of quadratic objectives. 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.