# Combinatorial persistency criteria for multicut and max-cut

## TL;DR

This paper introduces new criteria and algorithms for identifying partial optimal solutions in multicut and max-cut problems, enhancing preprocessing and solution quality in various applications.

## Contribution

It develops novel persistency criteria and efficient algorithms for verifying partial optimality in multicut and max-cut problems, applicable in practical scenarios.

## Key findings

- Effective in reducing problem sizes through preprocessing.
- Able to compute partial optimality guarantees for heuristic solutions.
- Demonstrated success on real-world instances from multiple domains.

## Abstract

In combinatorial optimization, partial variable assignments are called persistent if they agree with some optimal solution. We propose persistency criteria for the multicut and max-cut problem as well as fast combinatorial routines to verify them. The criteria that we derive are based on mappings that improve feasible multicuts, respectively cuts. Our elementary criteria can be checked enumeratively. The more advanced ones rely on fast algorithms for upper and lower bounds for the respective cut problems and max-flow techniques for auxiliary min-cut problems. Our methods can be used as a preprocessing technique for reducing problem sizes or for computing partial optimality guarantees for solutions output by heuristic solvers. We show the efficacy of our methods on instances of both problems from computer vision, biomedical image analysis and statistical physics.