ℓp-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 1.5 n log 0.5 k ) -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.