Optimality-Preserving Data Reduction for Maximum k-Cut (Full Version)

2026-07-03

AI Summary

This work addresses the limitation of existing exact solvers for large-scale Maximum k-Cut problems (k > 2), which stems from the absence of effective preprocessing techniques. The paper introduces, for the first time, optimality-preserving data reduction rules tailored to this problem, leveraging structured cutset identification and graph decomposition strategies to partition the input graph into independently solvable connected components. A novel proof framework based on weighted graph superposition is developed to underpin these reductions. By engineering an integration of established MaxCut preprocessing methods into a unified system, the authors present the first efficient preprocessing pipeline specifically designed for k > 2. Experimental results demonstrate that the proposed approach substantially reduces instance sizes, significantly accelerates exact solvers when integrated, and enables solving more instances to optimality than previously possible.

Abstract

Preprocessing has become an increasingly important part of solving Maximum Cut to optimality, enabling exact solvers to tackle significantly larger instances. This suggests that exact solvers for the more general Maximum k-Cut problem could also benefit from sophisticated preprocessing. However, to the best of our knowledge, no preprocessing techniques that are effective for k > 2 have been published. In this paper, we introduce structured cut sets, a novel data reduction technique for Maximum k-Cut. We provide criteria under which deleting cut sets is optimality-preserving, yielding a decomposition into connected components that can be solved independently and whose solutions can be combined into an optimal solution for the original graph. Furthermore, we extend several preprocessing techniques from Maximum Cut to Maximum k-Cut. To show that our rules are optimality-preserving, we develop a new proof framework based on the addition of weighted graphs. We complement our theoretical results by engineering a preprocessing framework for Maximum k-Cut and show its effectiveness in a computational study. The preprocessed instances are typically significantly smaller. Integrating our preprocessing into an exact solver yields significant speed-ups and enables solving more instances to optimality.

Problem

• Maximum k-Cut
• preprocessing
• data reduction
• optimality-preserving
• exact solvers

Innovation

• structured cut sets
• optimality-preserving reduction
• Maximum k-Cut
• preprocessing
• graph decomposition