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

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.