ℓ_p -Norm Multiway Cut

We introduce and study ℓ_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 ℓ_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= ∞), both of which are well-studied classic problems in the graph partitioning literature. We show that ℓ_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(log ^1.5nlog ^0.5k) -approximation for all p≥ 1. We also show an integrality gap of Ω (k^1-1/p) for a natural convex program and an O(k^1-1/p-ϵ) -inapproximability for any constant ϵ> 0 assuming the small set expansion hypothesis.