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 = infinity), 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 Omega(k(1-1/p)) for a natural convex program and an O(k(1-1)(/p-epsilon))-inapproximability for any constant epsilon > 0 assuming the small set expansion hypothesis.