# Combinatorial persistency criteria for multicut and max-cut

arXiv: https://arxiv.org/abs/1812.01426 · HTML: https://mlsanity.com/paper/1812.01426/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2018.

## Built on by (most cited first, 6 of 6)

- [RAMA: A Rapid Multicut Algorithm on GPU](https://mlsanity.com/paper/2109.01838/) (2021)
- [Structured Prediction Problem Archive](https://mlsanity.com/paper/2202.03574/) (2022)
- [Partial Optimality in Cubic Correlation Clustering](https://mlsanity.com/paper/2302.04694/) (2023)
- [An Unsupervised Learning Framework Combined with Heuristics for the Maximum Minimal Cut Problem](https://mlsanity.com/paper/2408.08484/) (2024)
- [Partial Optimality in Cubic Correlation Clustering for General Graphs](https://mlsanity.com/paper/2510.20431/) (2025)
- [Partial Optimality in the Preordering Problem](https://mlsanity.com/paper/2602.17346/) (2026)

## Abstract (arXiv)

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.