๐_p-Norm Multiway Cut
Karthekeyan Chandrasekaran, Weihang Wang ยท DROPS (Schloss Dagstuhl โ Leibniz Center for Informatics) ยท 2021
We introduce and study ๐_p-norm-multiway-cut: the input here is an undirected graph with non-negative edge weights along with k terminals and the goal is to find a partition of the vertex set into k parts each containing exactly one terminal so as to minimize the ๐_p-norm of the cut values of the parts. This is a unified generalization of min-sum multiway cut (when p = 1) and min-max multiway cut (when p = โ), both of which are well-studied classic problems in the graph partitioning literature. We show that ๐_p-norm-multiway-cut is NP-hard for constant number of terminals and is NP-hard in planar graphs. On the algorithmic side, we design an O(logยฒ n)-approximation for all p โฅ 1. We also show an integrality gap of ฮฉ(k^{1-1/p}) for a natural convex program and an O(k^{1-1/p-ฮต})-inapproximability for any constant ฮต > 0 assuming the small set expansion hypothesis.