# Memory Efficient Max Flow for Multi-label Submodular MRFs

## TL;DR

This paper introduces a memory-efficient max-flow algorithm that enables the optimal solution of large-scale multi-label submodular MRFs on standard computers, overcoming previous memory limitations.

## Contribution

A novel max-flow algorithm variant that significantly reduces memory usage for multi-label submodular MRFs, enabling practical large-scale applications.

## Key findings

- Reduces memory requirements from quadratic to linear in the number of labels.
- Allows solving large multi-label submodular MRFs on standard hardware.
- Maintains optimality of solutions despite memory efficiency improvements.

## Abstract

Multi-label submodular Markov Random Fields (MRFs) have been shown to be solvable using max-flow based on an encoding of the labels proposed by Ishikawa, in which each variable $X_i$ is represented by $\ell$ nodes (where $\ell$ is the number of labels) arranged in a column. However, this method in general requires $2\,\ell^2$ edges for each pair of neighbouring variables. This makes it inapplicable to realistic problems with many variables and labels, due to excessive memory requirement. In this paper, we introduce a variant of the max-flow algorithm that requires much less storage. Consequently, our algorithm makes it possible to optimally solve multi-label submodular problems involving large numbers of variables and labels on a standard computer.