# Engineering Kernelization for Maximum Cut

## TL;DR

This paper develops and tests new kernelization data reduction rules for the Max Cut problem, significantly improving solver performance on benchmark instances and enabling solutions to previously unsolvable large networks.

## Contribution

The authors engineer a comprehensive set of efficient kernelization rules for Max Cut and demonstrate their practical effectiveness on diverse benchmark datasets.

## Key findings

- Speedups of up to multiple orders of magnitude in solver runtimes.
- Successfully solved four previously unsolvable instances within a 10-hour limit.
- Significant improvements on synthetic, VLSI, image segmentation, social, and biological network datasets.

## Abstract

Kernelization is a general theoretical framework for preprocessing instances of NP-hard problems into (generally smaller) instances with bounded size, via the repeated application of data reduction rules. For the fundamental Max Cut problem, kernelization algorithms are theoretically highly efficient for various parameterizations. However, the efficacy of these reduction rules in practice---to aid solving highly challenging benchmark instances to optimality---remains entirely unexplored. We engineer a new suite of efficient data reduction rules that subsume most of the previously published rules, and demonstrate their significant impact on benchmark data sets, including synthetic instances, and data sets from the VLSI and image segmentation application domains. Our experiments reveal that current state-of-the-art solvers can be sped up by up to multiple orders of magnitude when combined with our data reduction rules. On social and biological networks in particular, kernelization enables us to solve four instances that were previously unsolved in a ten-hour time limit with state-of-the-art solvers; three of these instances are now solved in less than two seconds.