A Specialized Simplex Algorithm for Budget-Constrained Total Variation-Regularized Problems

TL;DR

This paper introduces a specialized simplex algorithm tailored for linear programs with total variation regularization and budget constraints, leveraging graph structures for efficiency.

Contribution

It characterizes basic solutions via rooted spanning forests and develops an accelerated simplex method based on this structure.

Findings

Achieves up to tenfold speed improvements over existing solvers

Provides a novel graph-based interpretation of the simplex method

Demonstrates practical efficiency on budget-constrained TV-regularized problems

Abstract

We consider a class of linear programs on graphs with total variation regularization and a budgetary constraint. For these programs, we give a characterization of basic solutions in terms of rooted spanning forests with orientation on the underlying graph. This leads to an interpretation of the simplex method in terms of simple graph operations on these underlying forests. We exploit this structure to produce an accelerated simplex method and empirically show that such improvements can lead to an order of magnitude improvement in time when compared to state-of-the-art solvers.