$\ell _p$-Norm Multiway Cut

Abstract

We introduce and study $\ell _p$-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 $\ell _p$-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 $p=\infty$), both of which are well-studied classic problems in the graph partitioning literature. We show that $\ell _p$-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 $p\ge 1$. 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.