Augmentation Algorithms for Integer Programs with Total Variation-like Regularization

TL;DR

This paper develops augmentation algorithms for solving integer programs with total variation-like regularization, utilizing the Graver basis for exact and heuristic solutions, and demonstrates competitive performance against existing solvers.

Contribution

It introduces a novel use of the Graver basis for integer programs with total variation regularization, including an exact algorithm and a randomized heuristic for constrained problems.

Findings

The randomized heuristic often outperforms state-of-the-art solvers.

The Graver basis can be effectively used for global optimization in this context.

Experimental results validate the efficiency of the proposed algorithms.

Abstract

We address a class of integer optimization programs with a total variation-like regularizer and convex, separable constraints on a graph. Our approach makes use of the Graver basis, an optimality certificate for integer programs, which we characterize as corresponding to the collection of induced connected subgraphs of our graph. We demonstrate how to use this basis to craft an exact global optimization algorithm for the unconstrained problem recovering a method first shown by Kolmogorov and Shioura in 2009. We then address the problem with an additional budget constraint with a randomized heuristic algorithm that samples improving moves from the Graver basis in a randomized variant of the simplex algorithm. Through comprehensive experiments, we demonstrate that this randomized algorithm is competitive with and often outperforms state-of-the-art integer program solvers.