Convex Submodular Minimization with Indicator Variables

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.

This paper was submitted as a seperate new submission by mistake. It is intended as a revised and extended version of arXiv:2209.13161 (titled "On polynomial-time solvability of combinatorial Markov random fields"). A revised version will be submitted there.